Chapter 1

Analytic Geometry in the Plane

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Section 1.1Vectors, Projections, and Coordinates in the Plane. Simplest Applications[38/84%]

Problem 1

Points $A(2,5)$ and $B(-3,2)$ are given. Find the projections of the vector $\overrightarrow {AB}$ on the coordinate axes.

Problem 2

Points $A(1,2)$ and $B(5,-1)$ are given. Find the angles that the vector $\overrightarrow {AB}$ makes with the axes $Ox$ and $Oy$ , as well as the length of this vector.

Problem 3

Points $A(2,-1)$ , $B(5,3)$ , $C(3,5)$ , $D(-5,11)$ are given. Find the angle between the vectors $\overrightarrow {AB}$ and $\overrightarrow {CD}$ .

Problem 4

Points $A(2,-1)$ , $B(-1,3)$ , $C(4,7)$ , $D(-1,-5)$ are given. Find the projection of the vector $\overrightarrow {AB}$ onto the direction of the vector $\overrightarrow {CD}$ .

Problem 5

Points $A(3,5)$ , $B(6,-2)$ are given. Find the projection of the vector $\overrightarrow {AB}$ on the axis directed from the origin along the bisector of the first coordinate angle.

Problem 6

Vectors are drawn from the origin to the points $A(1,2)$ , $B(-2,3)$ , $C(6,-10)$ . Find their geometric sum in magnitude and direction.

Problem 7

The opposite vertices of a rectangle are $A(3,7)$ , $B(11,-1)$ . Find the center of the rectangle.

Problem 8

From the point $A(2,3)$ a segment is drawn to the point $B(7,-2)$ and extended by the same length again. Find the coordinates of the end of the extension.

Problem 9

The segment $\overrightarrow {AB}$ is divided into three equal parts by the points $M_{1}(1,2)$ and $M_{2}(2,4)$ . Find the points $A$ and $B$ .

Problem 10

The points $M_{1}(1,1)$ , $M_{2}(2,2)$ , $M_{3}(3,-1)$ are three consecutive vertices of a parallelogram. Find the fourth vertex.

Problem 11

The midpoints of the sides of a triangle are the points $M_{1}(-2,1)$ , $M_{2}(2,3)$ , $M_{3}(4,-1)$ . Find the coordinates of the vertices.

Problem 12

Two consecutive vertices of a square are the points $A(2,3)$ and $B(6,6)$ . Where are the other vertices?

Problem 13

Two consecutive vertices of a regular hexagon are the points $M_{1}(0,0)$ and $M_{2}(4,0)$ . Where is the next vertex?

Problem 14

Masses $2$ and $1$ are placed at the points $(3,5)$ and $(9,-7)$ . Where is the center of mass of these masses?

Problem 15

Masses $m_{1}$ , $m_{2}$ and $m_{3}$ are placed at the points $M_{1}(x_{1},y_{1})$ , $M_{2}(x_{2},y_{2})$ , $M_{3}(x_{3},y_{3})$ respectively. Prove that the coordinates of the center of mass of these masses are given by the formulas

x_{c} = \frac{m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}}{m_{1}+m_{2}+m_{3}},

y_{c} = \frac{m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}.

Problem 16

Masses of equal magnitude are placed at the vertices of a polygon. Prove that the coordinates of their center of mass are equal to the arithmetic means of the coordinates of the vertices.

Problem 17

The center of a regular polygon is at the origin. Prove that the sum of the coordinates of the vertices equals zero.

Problem 18

At the sides of a polygon, perpendiculars are erected that are proportional to the sides and directed outward. Prove that their geometric sum equals zero.

Problem 19

The coordinates of the midpoints of the sides of a polygon with an odd number of sides are given: $M_{1}(x_{1},y_{1})$ , $M_{2}(x_{2},y_{2})$ , , $M_{2n-1}(x_{2n-1},y_{2n-1})$ . Find the coordinates of the vertices.

Problem 20

On the axis $Ox$ find the point whose distance to the point $(5,12)$ equals 13.

Problem 21

$M_{1}(2,1)$ , $M_{2}(-3,2)$ , $M_{3}(-1,1)$ are the vertices of a triangle. Find the center and radius of the circumscribed circle.

Problem 22

The vertices of a triangle are the points $O(0,0)$ , $M_{1}(3,5)$ , $M_{2}(-2,3)$ . Find its area.

Problem 23

The vertices of a triangle are the points $A(1,2)$ , $B(3,-1)$ , $C(-2,-5)$ . Find its area.

Problem 24

Two vertices of a triangle are at the points $(5,1)$ , $(-2,2)$ , and the third vertex is on the axis $Ox$ . Find it, given that the area equals 10.

Problem 25

Find the area of a quadrilateral from the coordinates of its vertices: $M_{1}(5,6)$ , $M_{2}(5,-6)$ , $M_{3}(-2,-1)$ , $M_{4}(-2,1)$ .

Problem 26

Find the area of a quadrilateral from the coordinates of its vertices: $M_{1}(5,6)$ , $M_{2}(5,-6)$ , $M_{3}(-2,1)$ , $M_{4}(-2,-1)$ .

Problem 27

Find the area of a pentagon from its vertices: $M_{1}(0,0)$ , $M_{2}(3,-2)$ , $M_{3}(5,-1)$ , $M_{4}(8,4)$ , $M_{5}(4,5)$ .

Problem 28

After a translation of the origin without rotation of the axes, the point $(2,4)$ acquired the coordinates $(-3,0)$ . Find the former coordinates of the new origin.

Problem 29

The new axes bisect the angles between the old ones. The axis $Ox_{1}$ makes a positive acute angle with $Oy$ . Set up the formulas for the coordinate transformation.

Problem 30

The new origin is at the point $(2,3)$ . The point $(6,0)$ is on the positive direction of the new ordinate axis. What are the new coordinates of the point $(7,8)$ ?

Problem 31

The vertices of a square are at the points $(0,0)$ , $(2,0)$ , $(2,2)$ , $(0,2)$ . Set up the formulas for the coordinate transformation if the diagonals of the square are taken as the new axes, and the point $(2,0)$ lies on the positive direction of the axis $O_{1}x_{1}$ .

Problem 32

The new origin is at the point $(1,-2)$ . The new ordinate axis makes with the old abscissa axis an acute angle whose tangent equals $3/4$ . Find the point whose old coordinates equal its new coordinates.

Problem 33

For what rotation of the axes does the quantity $x^{2}-y^{2}$ become $2x_{1}y_{1}$ ?

Problem 34

Prove that under a general coordinate transformation (with translation and rotation of the axes), the old axes can be brought into coincidence with the new ones by a rotation of the whole plane about some point.

Problem 35

The polar coordinates of a point are given: $r = 10$ , $\varphi = 30^{\circ }$ . Find its rectangular coordinates if the pole is at the point $(2,3)$ and the polar axis is parallel to the axis $Ox$ .

Problem 36

Find the distance between two points, given their polar coordinates: $r_{1}=3$ , $\varphi_{1}=10^{\circ }$ , $r_{2}=5$ , $\varphi_{2}=130^{\circ }$ .

Problem 37

The pole is at the point $(3,5)$ . The polar axis is parallel to the positive direction of the axis $Oy$ . Find the polar coordinates of the points $M_{1}(9,-1)$ and $M_{2}(5,5+2\sqrt{3})$ .

Problem 38

By considering the projections of a certain broken line, prove the formulas:

\sin \varphi + \sin 3\varphi + \ldots + \sin (2n-1)\varphi = \frac{1-\cos 2n\varphi }{2\sin \varphi };

\cos \varphi + \cos 3\varphi + \ldots + \cos (2n-1)\varphi = \frac{\sin 2n\varphi }{2\sin \varphi }.

Section 1.2The Line and the Circle[91/86%]

Problem 39

Find the equation of the line parallel to the axis $Ox$ and at distance $h$ from it.

Problem 40

Find the equation of the line parallel to the axis $Oy$ and at distance $h$ from it.

Problem 41

The vertices of a square are at the points $(0,0)$ , $(1,0)$ , $(1,1)$ , $(0,1)$ . Find the equations of its diagonals.

Problem 42

On the line $y = 2x-3$ find the point whose ordinate equals 7.

Problem 43

Assuming that the axis $Ox$ is horizontal, determine which of the points $M_{1}(-1,2)$ , $M_{2}(-3,-10)$ , $M_{3}(2,1)$ , $M_{4}(5,4)$ lie above, below, and on the line $y = 2x-3$ .

Problem 44

Find the equation of the line passing through the point $(2,1)$ and making an angle of $45^{\circ }$ with $Ox$ .

Problem 45

Draw the line through the points $M_{1}(-1,2)$ and $M_{2}(2,1)$ .

Problem 46

Find the equation of the line passing through the points $(3,7)$ and $(3,-2)$ .

Problem 47

Find the equations of the sides of the triangle whose vertices are at the points $M_{1}(1,-1)$ , $M_{2}(3,5)$ , $M_{3}(-7,11)$ .

Problem 48

Through the point $(2,-1)$ draw the line parallel to the line $2x+3y=0$ .

Problem 49

Through the point $(2,-3)$ draw the line perpendicular to the line $y = 2x+1$ .

Problem 50

Through the point $(3,5)$ draw lines at an angle of $45^{\circ }$ to the line $3x-2y+7=0$ .

Problem 51

Find the angles of the triangle with vertices at the points $M_{1}(0,2)$ , $M_{2}(2,2)$ , $M_{3}(3+\sqrt{3},3+\sqrt{3})$ .

Problem 52

Through the point $(4,-3)$ draw a line such that the area bounded by it and the coordinate axes equals 3 square units.

Problem 53

Find the vertices of the triangle from the equations of its sides: $x-y=0$ , $2x+3y+5=0$ , $x+2y+6=0$ .

Problem 54

Find the area of the triangle from the equations of its sides: $y = 3x-9$ , $y = -2x+1$ , $y = -x+3$ .

Problem 55

Through the point $(1,2)$ draw a line making an angle of $30^{\circ }$ with the line $x-2y=0$ , and find the point of intersection of the given line and the sought one.

Problem 56

A vertex of a square is given at $(2,-2)$ , and its diagonal is $x+2y=0$ . Find the equations of the sides of the square.

Problem 57

The line $3x-2y+2=0$ and the point $M(-4,8)$ are given. Find the projection $M'$ of the point $M$ onto the given line and determine the ratio $\left|OM'\right| : \left|OM\right|$ .

Problem 58

Through the point $(1,2)$ draw a line such that its distances to the points $(2,3)$ and $(4,-5)$ are equal.

Problem 59

The equations of two sides of a parallelogram are: $x+2y+1=0$ and $2x+y-3=0$ . Its center is at the point $(1,2)$ . Find the equations of the two other sides.

Problem 60

Through the point $(-1,2)$ draw a line whose distance to the point $(6,1)$ equals 5.

Problem 61

Through the point $(2,-2)$ draw lines whose distances to the point $(5,2)$ equal 3.

Problem 62

The point $M(-1,2)$ and the line $y=3x$ are given. Draw a line parallel to the given one, whose distance to the point $M$ is three times as large as its distance to the given line.

Problem 63

Find the equation of the line passing through the point $(-4,3)$ and at a distance from the origin equal to 5.

Problem 64

The line $4x+3y+1=0$ is given. Find the line parallel to the given one and at a distance of 3 from it.

Problem 65

Find the distance between the lines $2x+3y=7$ and $4x+6y=11$ .

Problem 66

Find the line parallel to the lines $x+2y=1$ and $x+2y=3$ , lying between them, and dividing the distance between them in the ratio $1:3$ .

Problem 67

Find the equation of the line lying midway between the lines $3x+2y=5$ and $6x+4y+3=0$ .

Problem 68

Find the equation of the line parallel to the line $2x+3y+6=0$ and cutting off from the coordinate angle a triangle of area 3.

Problem 69

The line $2x+y-3=0$ and the point $M(1,1)$ on it are given. On the same line find the points at distance $\sqrt{5}$ from $M$ .

Problem 70

Find the equations of the sides of the triangle for which $2x+3y+1=0$ and $x+y=0$ are altitudes, and $M(1,2)$ is one of the vertices.

Problem 71

$M_{1}(2,1)$ and $M_{2}(4,9)$ are vertices of a triangle, and $N(3,4)$ is the point of intersection of the altitudes. Find the equations of the sides.

Problem 72

The midpoints of the sides of a triangle are at the points $(1,2)$ , $(7,4)$ , $(3,-4)$ . Find the equations of the sides.

Problem 73

Two sides of a parallelogram $2y-x=0$ , $x-3=0$ and a vertex $(7,5)$ are given. Find the angle between the diagonals.

Problem 74

Two sides of a parallelogram $2x-y=0$ , $x-3y=0$ and its center $(2,3)$ are given. Find the equations of the diagonals.

Problem 75

Set up the equations of the sides of a triangle, knowing its vertex $(-4,3)$ and the equations of two medians: $3x-2y+3=0$ , $3x-5y+6=0$ .

Problem 76

The intersection of the medians is at the point $(-1,0)$ , and $x+y-1=0$ and $y+1=0$ are the equations of two sides. Find the equation of the third side.

Problem 77

The vertices of a triangle are at the points $A(1,3)$ , $B(-1,0)$ , $C(2,-2)$ . Find the equations of the altitudes.

Problem 78

Through the point of intersection of the lines $x-y-1=0$ and $x+2y-2=0$ draw the line passing through the point $(-1,1)$ .

Problem 79

Draw the line through the origin and the point of intersection of the lines $17x+29y=317$ and $3x+10y=634$ .

Problem 80

Through the point of intersection of the lines $x+2y-11=0$ and $2x-y-2=0$ draw the line whose distance from the origin equals 5.

Problem 81

Through the point $(-1,1)$ draw a line such that the midpoint of its segment between the lines $x+2y-1=0$ and $x+2y-3=0$ lies on the line $x-y-1=0$ .

Problem 82

Through the origin draw lines such that their segments between the lines $x-y+1=0$ and $x-y-2=0$ equal 3.

Problem 83

Find the line passing through the point $(2,3)$ , knowing that the segment of this line between the lines $3x+4y-7=0$ and $3x+4y+8=0$ equals $3\sqrt{2}$ .

Problem 84

Find the bisectors of the angles between the lines $3x+4y-1=0$ and $4x-3y+5=0$ .

Problem 85

Find the bisector of the angle between the lines $4x+7y-3=0$ and $8x-y+3=0$ in which the origin lies.

Problem 86

The points $(1,2)$ , $(-1,-1)$ , $(2,1)$ are the vertices of a triangle. Find the equation of the bisector of the interior angle at the point $(-1,-1)$ .

Problem 87

Find the radius of the circle inscribed in the triangle, given the equations of the sides: $3x-4y=25$ , $5x+12y=65$ , $8x+15y+85=0$ .

Problem 88

The sides of a triangle are $2x+y=0$ , $x-2y=0$ and $2x+11y-20=0$ . Find the center of the inscribed circle.

Problem 89

The point $P(0,1)$ is given. Draw a line through it such that its segment between the lines $x-3y+10=0$ and $2x+y-8=0$ is bisected by the point $P$ itself.

Problem 90

One side of an angle, $y=2x$ , and its bisector, $2y-x-3=0$ , are given. Find the other side of the angle.

Problem 91

Two vertices of a triangle, $A(1,6)$ and $B(7,-2)$ , and the center of the inscribed circle, $O(4,4)$ , are given. Find the third vertex $C$ .

Problem 92

The equations of the legs of an isosceles triangle are $2x-y+8=0$ and $x-2y-12=0$ . The point $(4,0)$ is on the base. Find the equation of the base.

Problem 93

A ray of light passes through the point $(2,3)$ , is reflected in the line $x+y+1=0$ , and arrives at the point $(1,1)$ . Find the equations of the incident ray and the reflected ray.

Problem 94

The line $2x+y-1=0$ is a bisector, and the points $(1,2)$ and $(-1,-1)$ are vertices of a triangle. Find the third vertex.

Problem 95

The point $(2,5)$ is a vertex of a triangle, and the lines $3x+4y-12=0$ and $x-y-1=0$ are the bisectors of the interior angles. Find the equations of the sides.

Problem 96

The points $(1,1)$ and $(0,4)$ are vertices of a triangle, and $2x-y-1=0$ is an interior bisector. Find the equations of the sides, given that the area of the triangle equals 5.

Problem 97

The equation of the base of an isosceles triangle is $x+y-1=0$ , and the equation of a leg is $x-2y-2=0$ . The point $(-2,0)$ is on the other leg. Find the equation of that leg.

Problem 98

The bisectors of the interior angles $A$ and $B$ of a triangle meet its sides at the points $M$ and $N$ . Prove that for any point of the segment $MN$ , the sum of the distances to the sides $AC$ and $BC$ equals the distance to the side $AB$ .

Problem 99

Through a point on the bisector of an angle two lines are drawn. One of them cuts off on the sides of the angle, measured from the vertex, segments $a$ and $b$ ; the other cuts off segments $a_{1}$ and $b_{1}$ . Prove that

\frac{1}{a}+\frac{1}{b} = \frac{1}{a_{1}}+\frac{1}{b_{1}}.

Problem 100

Take the lines $2x-y+1=0$ and $x+2y-1=0$ as the new axes $O_{1}x_{1}$ and $O_{1}y_{1}$ , choosing the directions so that the new coordinates of the old origin are positive. Find the formulas for passing from one system of coordinates to the other.

Problem 101

The lines $x-y-1=0$ and $x+y+2=0$ are taken as the new coordinate axes so that the new coordinates of the old origin are negative. Find the new equations of the old axes.

Problem 102

A line meets the sides $AB$ and $CB$ of a triangle at the points $M$ and $N$ , and the extension of the side $AC$ at the point $P$ . Prove the equality

AM \cdot BN \cdot CP = AP \cdot BM \cdot CN

(Menelaus's theorem).

Problem 103

The points $M$ , $N$ , $P$ lie on the sides $BC$ , $CA$ and $AB$ of a triangle. The lines $AM$ , $BN$ and $CP$ meet at one point. Prove the equality

AN \cdot BP \cdot CM = NC \cdot PA \cdot MB.

Problem 104

Prove that the equations of the bisectors of a triangle can be written in the form $s_{1}\pm s_{2}=0$ , $s_{1}\pm s_{3}=0$ , $s_{2}\pm s_{3}=0$ , where $s_{1}$ , $s_{2}$ and $s_{3}$ denote the left-hand sides of the equations of the sides of the triangle, taken in normal form.

Problem 105

A homographic, or projective, transformation of the plane is one in which each point $(x,y)$ passes to the point $(x_{1},y_{1})$ , where $x_{1} = \frac{a_{1}x+b_{1}y+c_{1}}{a_{0}x+b_{0}y+c_{0}}$ ; $y_{1} = \frac{a_{2}x+b_{2}y+c_{2}}{a_{0}x+b_{0}y+c_{0}}$ ; $a_{0}^{2}+b_{0}^{2}+c_{0}^{2} > 0$ .

Prove that for any choice of the coefficients $a_{k}$ , $b_{k}$ , $c_{k}$ ( $k=0,1,2$ ), the projective transformation represents a collineation, i.e., it carries lines into lines.

Problem 106

The cross-ratio (anharmonic ratio) of four points $A$ , $B$ , $C$ , $D$ lying on a line is defined as the ratio $\frac{AC}{BC} : \frac{AD}{BD}$ . Prove that under a projective transformation the value of the cross-ratio does not change.

Problem 107

Find the general form of the equations of circles tangent to the axis $Ox$ at the origin.

Problem 108

Find the equation of the circles of the preceding problem in polar coordinates, if the polar axis coincides with the positive part of the axis $Ox$ .

Problem 109

Find the center and radius of the circle $x^{2}+y^{2}-6x+8y=0$ .

Problem 110

Set up the equation of a circle of radius $r=2$ , tangent to the lines $3x+4y-8=0$ and $12x-5y+17=0$ .

Problem 111

Find the length of the segment of the line $x+5y=9$ lying inside the circle $x^{2}+y^{2}-4x-8y+7=0$ .

Problem 112

Find the equation of the circle circumscribed about the triangle with vertices at the points $M_{1}(0,0)$ , $M_{2}(10,0)$ , $M_{3}(6,8)$ .

Problem 113

On the circle $x^{2}+y^{2}=1$ find the point equidistant from the points $(1,3)$ and $(-2,2)$ .

Problem 114

Find the equation of the circle inscribed in the triangle whose sides are given by the equations:

3x+4y=25, \qquad 5x-12y=65, \qquad 8x-15y+85=0.

Problem 115

Through the point $(1,2)$ draw a tangent to the circle $x^{2}+y^{2}=5$ .

Problem 116

Through the point $(-1,3)$ draw tangents to the circle $x^{2}+y^{2}=5$ .

Problem 117

Find the common chord of the circles $x^{2}+y^{2}=2ax$ and $x^{2}+y^{2}=2by$ .

Problem 118

Find the common tangents to the circles $x^{2}+y^{2}=6x$ and $x^{2}+y^{2}=6y$ .

Problem 119

The point $(x_{1},y_{1})$ is outside the circle $x^{2}+y^{2}+Ax+By+C=0$ . Find the length $l$ of the tangent from the point to the circle.

Problem 120

The point $(x_{1},y_{1})$ is inside the circle $x^{2}+y^{2}+Ax+By+C=0$ . Prove that the chord of the circle passing through $(x_{1},y_{1})$ is divided by this point into parts whose product equals $x_{1}^{2}+y_{1}^{2}+Ax_{1}+By_{1}+C$ .

Problem 121

A circle is tangent to the coordinate axes and passes through the point $(4,8)$ . Find its equation.

Problem 122

A circle is tangent to the axis $Oy$ and passes through the origin and the point $(3,6)$ . Find its equation.

Problem 123

On the axis $Ox$ find a point such that the tangents from it to the circles $x^{2}+y^{2}=6y-6$ and $x^{2}+y^{2}=2x$ have equal length.

Problem 124

Find the general form of the circles tangent to both bisectors of the coordinate angles.

Problem 125

Show that the locus of points whose distances to two given points are in a given ratio not equal to unity is a circle. (Circle of Apollonius.)

Problem 126

Show that the locus of points from which the tangents to two given circles have equal length is a line. (Radical axis of two circles.)

Problem 127

Prove that for any $a$ and $b$ the circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=by$ intersect at a right angle. Here, as usual, the angle between the curves is understood as the angle between their tangents at the point of intersection.

Problem 128

Show that each of the circles $(x-a)^{2}+y^{2}=a^{2}+h^{2}$ intersects each of the circles $x^{2}+\left(y\pm \sqrt{r^{2}+h^{2}}\right)^{2}=r^{2}$ at a right angle.

Problem 129

Prove that each of the circles $x^{2}+y^{2}-2ax+b^{2}=0$ intersects each of the circles $x^{2}+y^{2}-2cy+b^{2}=0$ at a right angle.

Section 1.3Loci[34/76%]

Problem 130

Chords of the circle $x^{2}+y^{2}=2ax$ are drawn from the origin. Find the locus of the midpoints of these chords.

Problem 131

Find the equation of the locus of points whose distances to the lines $Ax+By+C=0$ and $A_{1}x+B_{1}y+C_{1}=0$ are in the ratio $m:n$ .

Problem 132

A segment of length $a+b$ slides with its ends on the coordinate axes. Find the curve traced by the point $M$ dividing it into parts $a$ and $b$ . (Leonardo da Vinci's ellipsograph.)

Problem 133

Two circles of radii $b$ and $a$ are drawn centered at the origin. A variable radius meets the inner one at the point $A$ , and the outer one at $B$ . From $A$ a line is drawn parallel to the axis $Ox$ , and from $B$ a line parallel to the axis $Oy$ , up to their mutual intersection at the point $M$ . Find the locus of the point $M$ .

Problem 134

A line is drawn from the origin making an angle $\frac{\pi \theta }{2}$ with the axis $Oy$ . This line meets the line $x=a\theta$ at the point $M$ . Find the equation of the locus of the point $M$ . (Quadratrix of Dinostratus.)

Problem 135

Chords of the circle $x^{2}+y^{2}=2ax$ are drawn from the origin and extended to their intersection with the line $x=2a$ . These extensions are then laid off on the same chords starting from the origin. Find the locus of the endpoints of the shifted extensions. (Cissoid of Diocles.)

Problem 136

An arbitrary line is drawn from the origin, meeting the circle $x^{2}+y^{2}=ay$ and the line $y=a$ at the points $A$ and $B$ . From the point $A$ a line is drawn parallel to the axis $Ox$ , and from the point $B$ a line parallel to the axis $Oy$ . Find the locus of the point of intersection of these lines. (Witch of Agnesi.)

Problem 137

Find the locus of points the product of whose distances to two given points $(a,0)$ and $(-a,0)$ is a constant quantity equal to $b^{2}$ . (Ovals of Cassini.)

Problem 138

Find the locus of points the product of whose distances to two given points equals the square of half the distance between them. (Lemniscate of Bernoulli.)

Problem 139

Find the locus of points between whose distances to two given points there holds the linear relation $r_{1}-ar_{2}=b$ . (Ovals of Descartes.)

Problem 140

Lines are drawn through the point $(0,-a)$ meeting the axis $Ox$ . On each of them, on both sides of the point of intersection with $Ox$ , segments of length $h$ are laid off. Find the locus of the endpoints of these segments. (Conchoid of Nicomedes.)

Problem 141

From a point $A$ on a circle of diameter $a$ , secants are drawn. On each of them, on both sides of the other point of intersection with the circle, segments of length $b$ are laid off. Find the locus of their endpoints. (Limaçon of Pascal.)

Problem 142

A thread wound around the circle $x^{2}+y^{2}=a^{2}$ is unwound while remaining taut. Find the locus traced by its end, if the initial position of the end of the thread was at the point $(a,0)$ . (Involute of a circle.)

Problem 143

A circle of radius $a$ rolls without slipping along the axis $Ox$ . Find the equation of the curve traced by the point of the circle that, at the initial moment, touched the axis $Ox$ at the origin. (Cycloid.)

Problem 144

Find the equation of the curve traced by a point of a circle of radius $a$ rolling without slipping along the axis $Ox$ , if, at the moment when the circle touched the axis $Ox$ at the origin, this point occupied the position $(0,a-b)$ . (Trochoid.)

Problem 145

A circle of radius $a$ rolls without slipping on the outside of the circle $x^{2}+y^{2}=a^{2}n^{2}$ . Find the locus traced by the point of the rolling circle that, at the initial moment, touched the fixed circle on the axis $Ox$ . (Epicycloid.)

Problem 146

A circle of radius $a$ rolls without slipping inside the circle $x^{2}+y^{2}=a^{2}n^{2}$ . Find the locus traced by the point of the rolling circle that, at the initial moment, touched the fixed circle on the axis $Ox$ . (Hypocycloid.)

Problem 147

Show that for $n=4$ the hypocycloid becomes the astroid $x^{2/3}+y^{2/3}=(4a)^{2/3}$ .

Problem 148

For $n=1$ the epicycloid becomes a curve called the cardioid. Show that the equation of the cardioid, in suitably chosen polar coordinates, has the form $r = a(1+\cos \varphi )$ .

Problem 149

Show that the cardioid (see the preceding problem) is a special case of the limaçon of Pascal (see problem 141).

Problem 150

Show that for $n=2$ the hypocycloid becomes a line. (Cardano's theorem.)

Problem 151

A right triangle with legs $a$ and $b$ slides with its ends on the coordinate axes. Find the locus of the vertex of the right angle.

Problem 152

Find the locus of the centers of rectangles inscribed in a given triangle so that one of their sides lies on the base of the triangle.

Problem 153

Find the locus of the centers of parallelograms inscribed in a given quadrilateral, whose sides are parallel to the diagonals of the quadrilateral.

Problem 154

The endpoints of the base of a triangle are at the points $(\pm a,0)$ . One of the angles at the base is twice the other. Find the locus of the vertex of the triangle.

Problem 155

The endpoints of the base of a triangle are at the points $(\pm a,0)$ . The difference of the angles at the base equals $\varphi$ . Find the locus of the vertex.

Problem 156

Find the locus of the centroids of the triangles formed by the lines $x=0$ , $y=0$ , $\frac{x}{k}+\frac{y}{3-k}=1$ .

Problem 157

Three vertices of a parallelogram, the directions of whose sides are given, slide along three given lines. What is the locus of the fourth vertex?

Problem 158

The four sides of a changing parallelogram all pass, at all times, through four given points on a line. Show that the diagonals of the parallelogram also pass through certain fixed points.

Problem 159

The sides of a right angle, whose position changes, pass at all times through two given points. Prove that its bisector also passes through a fixed point.

Problem 160

A side of a variable square passes through the origin, and its endpoints slide along lines parallel to the axis $Ox$ . What is the locus of the other two vertices?

Problem 161

Find the locus of the midpoints of the chords of the circle $x^{2}+y^{2}=a^{2}$ passing through the point $P(c,0)$ inside the circle.

Problem 162

Lines $a$ , $b$ and $c$ rotate about the points $A$ , $B$ , $C$ lying on one line. As they do so, the point of intersection of the lines $a$ and $b$ slides along a line $P$ , and the point of intersection of the lines $a$ and $c$ slides along a line $Q$ . Prove that the locus of the points of intersection of the lines $b$ and $c$ is a line $R$ passing through the point of intersection of the lines $P$ and $Q$ .

Problem 163

Prove Desargues's theorem: if the vertices of two triangles lie on three lines issuing from one point, then the three points of intersection of the corresponding pairs of sides of the triangles lie on one line.

Section 1.4Second-Order Curves in Simplest Form[90/74%]

Problem 164

Find the equation of the parabola passing through the point $(6,9)$ , with vertex at the origin and symmetric with respect to the axis $Oy$ .

Problem 165

The axis $Ox$ is the axis of symmetry of a parabola with vertex at the origin. Find the equation of this parabola, given that it passes through the point $(2,2)$ .

Problem 166

The parabolic mirror of the telescope reflector at Simeiz had a focal length of $5{,}4$ m and a diameter of $1{,}02$ m. Find the depth of the parabolic concavity of the mirror.

Problem 167

The mirror of an automobile headlight has, in cross-section, the shape of a parabola. The diameter of the mirror is $20$ cm, its depth is $10$ cm. Find the position of the focus of the mirror.

Problem 168

Prove that all parabolas $y^{2}=2px$ are geometrically similar.

Problem 169

On the parabola $y^{2}=24x$ a point is taken at a distance of 14 units from the focus. Determine its distance from the vertex.

Problem 170

Prove that the product of the ordinates of the endpoints of any chord passing through the focus of the parabola $y^{2}=2px$ is a constant quantity.

Problem 171

On the parabola $y^{2}=2px$ find a point such that its distances from the focus and from the vertex are in the ratio $7:8$ .

Problem 172

The parabola $y^{2}=4-4x$ and the line $x+y=1$ are given. Through one of their points of intersection, draw another parabola with the same axis and focus.

Problem 173

Set up the equation of the parabola with axis of symmetry $Ox$ , passing through the points $(a,0)$ , $(0,b)$ . For what relation between $a$ and $b$ will the focus of the parabola be at the origin?

Problem 174

The parabolas $y^{2}=4x+8$ and $4x+y^{2}-4y-8=0$ are given. Find the equation of their common chord and its point of intersection with the line joining the foci.

Problem 175

Three points $(-5,2)$ , $(1,-4)$ and $(5,12)$ are given, lying on a parabola whose axis is parallel to the axis $Oy$ . Find the vertex, focus, and directrix of this parabola.

Problem 176

Through three points $(-2,1)$ , $(-1,0)$ and $(6,1)$ , draw a circle and a parabola with axis parallel to the axis $Oy$ . Determine the distance from the center of the circle to the vertex of the parabola.

Problem 177

Prove that the equation $yy_{1}=p(x+x_{1})$ represents the tangent to the parabola $y^{2}=2px$ at the point $(x_{1},y_{1})$ .

Problem 178

The parabolas $y^{2}+4x=0$ and $y^{2}=16x+80$ are given. Verify that they are confocal and that the tangents at the points of intersection are mutually perpendicular.

Problem 179

Three points $(-3,2)$ , $(-1,1)$ and $(3,5)$ are given; draw through them a parabola with axis parallel to the axis $Oy$ , and a circle, and find the angles at which they intersect.

Problem 180

Prove that the tangent to the parabola $y^{2}=2px$ having slope $m$ is represented by the equation $y = mx+\frac{p}{2m}$ .

Problem 181

Find the tangent to the parabola $4y=x^{2}$ at the point $(2,1)$ .

Problem 182

Through the point $(0,-4)$ draw a tangent to the parabola $4y=x^{2}$ .

Problem 183

Through the point $(-4,-1)$ draw a tangent to the parabola $y^{2}=2x$ .

Problem 184

Prove that the tangents drawn at the endpoints of a chord of a parabola passing through the focus are mutually perpendicular.

Problem 185

The parabola $y^{2}=2px$ is given. Find the angle between the tangents to it drawn from the point $(-p,p)$ .

Problem 186

Find the distance from the focus of the parabola $y^{2}=2px$ to a tangent to it whose angle with the axis equals $\alpha$ .

Problem 187

Find the locus of the feet of the perpendiculars dropped from the focus of a parabola onto its tangents.

Problem 188

The parabola $4y+x^{2}-4x-12=0$ and the line $y-2x=0$ are given. Draw the tangent parallel to the given line.

Problem 189

Find the common tangents to the parabola $y^{2}=8x$ and the circle $x^{2}+y^{2}=2$ .

Problem 190

Two parabolas are given: $y^{2}=2px$ and $y^{2}=2p(x+a)$ . Prove that the segments of the tangent to the first parabola lying inside the second are bisected by the point of tangency.

Problem 191

Find the locus of points from which the parabola is seen at a right angle.

Problem 192

The ordinates of the circle $x^{2}+y^{2}=36$ are decreased by half. Find the equation of the resulting curve.

Problem 193

Find the semi-axes of the ellipse $3x^{2}+5y^{2}-30=0$ .

Problem 194

Find the equation of the ellipse passing through the points $(1,4)$ and $(7,2)$ and symmetric with respect to the axes $Ox$ and $Oy$ .

Problem 195

Prove that ellipses with the same eccentricity $e$ are geometrically similar, i.e., can be brought into coincidence with each other by suitably enlarging one of them.

Problem 196

The meridian of the terrestrial globe is an ellipse whose flattening, i.e., $\frac{a-b}{a}$ , equals $\frac{1}{298}$ . Find its eccentricity.

Problem 197

The orbit of the terrestrial globe is an ellipse with semi-axis $a=150 \cdot 10^{6}$ km and eccentricity $e=0{,}017$ . Knowing that the Sun is at the focus of this ellipse, find by how much the shortest distance from the Earth to the Sun (January 4) is shorter than the longest (July 3).

Problem 198

Two points are given, lying on an ellipse whose axes are the coordinate axes — $(3;2{,}4)$ and $(4;1{,}8)$ . Find this ellipse and the projections of the foci onto the chord joining the given points.

Problem 199

An ellipse and a parabola are given: $y^{2}=18x-0{,}36x^{2}$ , $y^{2}=18x$ . Find the distance from the focus of the parabola to the nearest focus of the ellipse.

Problem 200

Find the eccentricity of an equilateral (rectangular) hyperbola.

Problem 201

The directrices of a hyperbola divide the distance between the foci into three equal parts. Find its eccentricity.

Problem 202

The foci of an ellipse divide the distance between the directrices into three equal parts. Find its eccentricity.

Problem 203

What will the equation of the equilateral hyperbola $x^{2}-y^{2}=a^{2}$ become if the axes are rotated through an angle $\alpha =-45^{\circ }$ ?

Problem 204

The distance between the foci of an ellipse equals 2, and the distance between the directrices equals 10. Find the semi-axes.

Problem 205

The eccentricity of a hyperbola equals 2. Find the angle between the asymptotes.

Problem 206

The minor axis of an ellipse is seen from the focus at a right angle. Find the eccentricity.

Problem 207

The hyperbola $x^{2}-y^{2}=8$ is given. Find the confocal ellipse passing through the point $(4,6)$ .

Problem 208

The ellipse $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$ is given. Write the equation of the confocal equilateral hyperbola.

Problem 209

Find the general equation of the ellipses and hyperbolas with foci at the points $(\pm c,0)$ .

Problem 210

The ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{5}=1$ is given. Find the equation of the hyperbola having its foci at the vertices of the given ellipse, and its vertices at the foci.

Problem 211

The ellipse $\frac{x^{2}}{33}+\frac{y^{2}}{8}=1$ is given. Find the hyperbola confocal with it, having eccentricity equal to $1{,}25$ .

Problem 212

Find the ellipse and the hyperbola passing through the point $(6,4)$ and confocal with the hyperbola

x^{2}-y^{2}+8=0.

Problem 213

The coordinate axes serve as the axes of a hyperbola passing through the point $(6,5)$ and having eccentricity equal to $1{,}5$ . Set up its equation.

Problem 214

Find the points at which the hyperbola $\frac{x^{2}}{64}-\frac{y^{2}}{36}=1$ is met by the lines parallel to the asymptotes and passing through the right focus.

Problem 215

Find the distance from the focus of the hyperbola $b^{2}x^{2}-a^{2}y^{2}=a^{2}b^{2}$ to an asymptote.

Problem 216

The polar equation of a second-order curve is $r(5+3\cos \varphi )=16$ . Find its equation with respect to its axes of symmetry.

Problem 217

The same, for the curve $r(4+5\cos \varphi )=9$ .

Problem 218

Prove the equality

\frac{1}{\rho _{1}}+\frac{1}{\rho _{2}}+\ldots +\frac{1}{\rho _{n}} = \frac{n}{p},

where $\rho_{1}$ , $\rho_{2}$ , , $\rho_{n}$ are the radius vectors drawn from the focus of a parabola at angles $\frac{2\pi }{n}$ to one another.

Problem 219

Prove the equality

\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}} = \frac{1}{a_{1}^{2}}+\frac{1}{b_{2}^{2}},

where $r_{1}$ and $r_{2}$ are mutually perpendicular radius vectors drawn from the center of an ellipse.

Problem 220

Prove the equality

\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\ldots +\frac{1}{r_{n}^{2}} = \frac{n}{2}\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}\right),

where $r_{1}$ , $r_{2}$ , , $r_{n}$ are radius vectors from the center of an ellipse, making angles $\frac{2\pi }{n}$ with one another.

Problem 221

Find the equations of the diameters of the ellipse $x^{2}+6y^{2}=2$ whose length equals 2.

Problem 222

Prove that under projection, mutually perpendicular diameters of a circle pass into conjugate diameters of an ellipse.

Problem 223

Using the theorem that the area of a projection equals the area of the figure itself multiplied by the cosine of the angle between the plane of the figure and the plane of projection, prove Apollonius's theorem: $a_{1}b_{1}\sin \omega = ab$ , where $a_{1}$ and $b_{1}$ are the lengths of conjugate semi-diameters of an ellipse, and $\omega$ is the angle between them.

Problem 224

Starting from the result of problem 222, prove another theorem of Apollonius, according to which $a_{1}^{2}+b_{1}^{2}=a^{2}+b^{2}$ .

Problem 225

Find the angle between the equal conjugate diameters of the ellipse $x^{2}+3y^{2}=6$ .

Problem 226

Find the lengths of the conjugate diameters of the hyperbola $x^{2}-y^{2}=1$ between which the angle equals $45^{\circ }$ .

Problem 227

Find the lengths of the conjugate diameters of the ellipse $x^{2}+15y^{2}=5$ between which the angle equals $150^{\circ }$ .

Problem 228

The angle between conjugate diameters of an ellipse equals $120^{\circ }$ . One of them is twice the other. Find the eccentricity.

Problem 229

Find the lengths of the conjugate diameters of the ellipse $\frac{x^{2}}{5}+3y^{2}=1$ making an angle of $150^{\circ }$ .

Problem 230

The lengths of the conjugate diameters of the ellipse $8x^{2}+17y^{2}=136$ are in the ratio $4:3$ . Find the equations of these diameters.

Problem 231

The sum of the lengths of conjugate diameters is 6, the angle between them is 150, and the eccentricity is $e=\frac{3}{5}$ . Find the axes of the ellipse.

Problem 232

Find the lengths of the conjugate diameters of the ellipse $3x^{2}+5y^{2}=15$ making the largest angle.

Problem 233

Find the angle between the conjugate diameters of a hyperbola, given the ratio $m$ of their lengths and the eccentricity $e$ .

Problem 234

Find the angle between the asymptotes of a hyperbola, given that its conjugate diameters form an angle of $45^{\circ }$ and are in the ratio 2 to 3.

Problem 235

Find the lengths of the conjugate diameters of the hyperbola $9x^{2}-16y^{2}=144$ whose sum is to the sum of the axes as 5 to 2.

Problem 236

Prove that the equation $Axx_{1}+Byy_{1}+C=0$ represents the tangent to the curve $Ax^{2}+By^{2}+C=0$ at the point $(x_{1},y_{1})$ .

Problem 237

The ellipse $4x^{2}+y^{2}=8$ and the line $y-2x=0$ are given. Draw the tangents to the ellipse parallel to this line, and find the distance between the tangents.

Problem 238

Through the point $(2,-1)$ draw a tangent to the ellipse $x^{2}+9y^{2}=9$ .

Problem 239

Through the point $(3,-6)$ draw a tangent to the hyperbola $4x^{2}-9y^{2}=36$ .

Problem 240

Through the point $(1,-2)$ draw a tangent to the hyperbola $x^{2}-y^{2}=1$ .

Problem 241

Draw, parallel to the line $2x-3y=0$ , tangents to the ellipse $x^{2}+4y^{2}=4$ .

Problem 242

Parallel to the line $10x+3y=0$ , draw tangents to the hyperbola $4x^{2}-y^{2}=4$ .

Problem 243

Find the tangents to the hyperbola $\frac{4x^{2}}{10}-\frac{y^{2}}{6}=1$ at its points of intersection with the line $3x=5y$ , and the distance between these tangents.

Problem 244

Given: the ellipse $\frac{x^{2}}{45}+\frac{y^{2}}{36}=1$ and the hyperbola $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$ . Verify that the tangents to them at the points of intersection are mutually perpendicular.

Problem 245

Two ellipses are given: $\frac{x^{2}}{15}+\frac{y^{2}}{9}=1$ and $\frac{x^{2}}{19}+\frac{y^{2}}{3}=1$ . Find the tangents to the first ellipse passing through the foci of the second.

Problem 246

Find the tangents to the hyperbola $\frac{x^{2}}{15}+\frac{y^{2}}{21}=1$ passing through a focus of the conjugate hyperbola.

Problem 247

Find the angle between the tangents to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ drawn from the point $(1,2)$ .

Problem 248

Prove that the tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ having slope $m$ is represented by the equation

y = mx \pm \sqrt{a^{2}m^{2}+b^{2}}.

Problem 249

Prove that the tangent to the hyperbola $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\sigma$ having slope $m$ is represented by the equation

y = mx \pm \sqrt{\sigma (a^{2}m^{2}-b^{2})}.

Problem 250

Prove that the chords joining the points of intersection of the circle $x^{2}+y^{2}=a^{2}+b^{2}$ with the coordinate axes are tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ .

Problem 251

Show that the ellipses $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ are tangent to the lines $\pm x \pm y = c$ , if $a^{2}+b^{2}=c^{2}$ .

Problem 252

Prove that the locus of points from which the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is seen at a right angle is the circle $x^{2}+y^{2}=a^{2}+b^{2}$ .

Problem 253

Show that for the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ , the product of the distances of the foci from a tangent equals $b$ .

Section 1.5Second-Order Curves Given by an Equation in General Form[17/71%]

Problem 254

Determine the type of the curves given by the second-order equations:

(a)

$xy=0$ ;

(b)

$x^{2}-y^{2}=0$ ;

(c)

$(x-y)^{2}=0$ ;

(d)

$(x-y)^{2}-3(x-y)+2=0$ ;

(e)

$x^{2}+y^{2}=0$ ;

(f)

$x^{2}+y^{2}+1=0$ .

Problem 255

(a)(b)(c)

Determine the type of the curves given by the second-order equations:

(a)

$3x^{2}-2xy+3y^{2}-2x+2y+1=0$ ;

(b)

$x^{2}+6xy+y^{2}+6x+2y-1=0$ ;

(c)

$x^{2}+2xy+y^{2}+2x-2y-1=0$ .

Problem 256

Determine $k$ so that the equation $x^{2}+6xy+y^{2}+6x+2y+k=0$ represents a pair of intersecting lines.

Problem 257

Determine $k$ so that the equation $3x^{2}-2xy+3y^{2}-2x+2y+k=0$ represents a point.

Problem 258

Choose $\lambda$ so that the equation $x^{2}-2\lambda xy+4y^{2}+2x+\lambda y=0$ represents a pair of lines, and find them.

Problem 259

In the equation $2x^{2}+\lambda xy+2y^{2}-7x+\mu y+3=0$ , choose the coefficients $\lambda$ and $\mu$ so that the equation represents a pair of parallel lines.

Problem 260

Find the equations of the asymptotes of the curve $2x^{2}-xy+3x+y-1=0$ .

Problem 261

Show that the equation $(ax+by+c)(a_{1}x+b_{1}y+c_{1})+\lambda =0$ , where $\lambda \neq 0$ and $ab_{1}-a_{1}b \neq 0$ , represents a hyperbola with asymptotes $ax+by+c=0$ , $a_{1}x+b_{1}y+c_{1}=0$ .

Problem 262

Find the equation of the hyperbola having as asymptotes the lines $x-1=0$ and $y-1=0$ and passing through the point $(2,2)$ .

Problem 263

Prove that for any values of the coefficients $\lambda$ and $\mu$ , the curve $\lambda f_{1}+\mu f_{2}=0$ passes through the points of intersection of the curves $f_{1}=0$ and $f_{2}=0$ . Here $f_{1}$ and $f_{2}$ are polynomials in $x$ and $y$ .

Problem 264

Let $s_{kl}=0$ be the equation of the line $A_{kl}x+B_{kl}y+C_{kl}=0$ passing through the points $k$ and $l$ . Prove that for any $\lambda$ and $\mu$ , the equation $\lambda s_{12}s_{34}+\mu s_{13}s_{24}=0$ represents a second-order curve passing through the four given points.

Problem 265

Find the equation of the second-order curve passing through the points $(1,1)$ , $(2,-1)$ , $(1,-2)$ , $(-1,1)$ , $(3,0)$ .

Problem 266

Find the second-order curve from five of its points: $(3,1)$ , $(2,1)$ , $(-7,1)$ , $(-2,0)$ , $(0,1)$ .

Problem 267

Find the parabola passing through the four points: $(0,-1)$ , $(0,3)$ , $(-1,0)$ , $(3,0)$ .

Problem 268

Find the second-order curve passing through the point $(1,1)$ and the points of intersection of the curve $x^{2}-2xy+2x-2y-17=0$ with the lines $y=0$ and $x+2y+3=0$ .

Problem 269

Through the origin, chords $AB$ and $CD$ of the circle $x^{2}+(y-h)^{2}=a^{2}$ are drawn. By considering the pencil of second-order curves passing through the endpoints of the chords, prove that the lines $AD$ and $CB$ , and likewise $BD$ and $AC$ , meet the axis $Ox$ at equal distances from the origin.

Problem 270

On a second-order curve, six points are taken: 1, 2, 3, 4, 5, 6. The equations of the sides of the resulting hexagon are denoted respectively as follows:

s_{12}=0, \quad s_{23}=0, \quad s_{34}=0, \quad s_{45}=0, \quad s_{56}=0, \quad s_{61}=0.

Under these conditions, and for any choice of the coefficients $\lambda$ and $\mu$ , the third-order curve $\lambda s_{12}s_{34}s_{56}+\mu s_{23}s_{45}s_{61}=0$ passes through the 6 given points and through the 3 points of intersection of the pairs of opposite sides.

Starting from this, prove Pascal's theorem: in a hexagon inscribed in a second-order curve, the three points of intersection of the pairs of opposite sides lie on one line.

Section 1.6Center, Diameters. Simplification of Equations of Second-Order Curves[33/67%]

Problem 271

Let $(x,y)$ be the midpoint of a chord of the curve $f(x,y,1)=0$ , and let $(x+\xi t, y+\eta t)$ and $(x-\xi t, y-\eta t)$ be the endpoints of the chord. Prove that

\left(\frac{\partial f}{\partial x}\right)_{1}\xi + \left(\frac{\partial f}{\partial y}\right)_{1}\eta = 0.

Remark.

The polynomial $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ , homogenized by introducing powers of the variable $z$ , becomes the ternary quadratic form $f = f(x,y,z) = Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}$ . Its partial derivatives with respect to $x$ , $y$ and $z$ are given by: $\frac{\partial f}{\partial x} = 2Ax+By+Dz$ ; $\frac{\partial f}{\partial y} = Bx+2Cy+Ez$ ; $\frac{\partial f}{\partial z} = Dx+Ey+2Fz$ . Their values at $z=1$ are denoted by $\left(\frac{\partial f}{\partial x}\right)_{1}$ , $\left(\frac{\partial f}{\partial y}\right)_{1}$ and $\left(\frac{\partial f}{\partial z}\right)_{1}$ . For brevity one also writes $f(x,y,1) = Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ , $f(x,y,0) = Ax^{2}+Bxy+Cy^{2}$ .

Problem 272

Prove that the midpoints of the chords of the curve $f(x,y,1)=0$ parallel to the line $\frac{x}{\xi } = \frac{y}{\eta }$ lie on the line $\left(\frac{\partial f}{\partial x}\right)_{1}\xi + \left(\frac{\partial f}{\partial y}\right)_{1}\eta = 0$ , or, in full form:

(2Ax+By+D)\xi + (Bx+2Cy+E)\eta = 0.

(The diameter conjugate to the given direction.)

Problem 273

Show that when the origin is translated to the point $(\xi ,\eta )$ , the equation $f(x,y,1)=0$ passes into the equation

f(x_{1},y_{1},0) + \frac{\partial f}{\partial \xi }x_{1} + \frac{\partial f}{\partial \eta }y_{1} + f(\xi ,\eta ,1) = 0.

Problem 274

If $4AC-B^{2} \neq 0$ , all the diameters pass through the point $(\xi ,\eta )$ where the lines $\left(\frac{\partial f}{\partial x}\right)_{1}=0$ , $\left(\frac{\partial f}{\partial y}\right)_{1}=0$ intersect. Show that when the origin is translated to the point chosen in this way, the equation $f(x,y,1)=0$ passes into the following one:

Ax_{1}^{2}+Bx_{1}y_{1}+Cy_{1}^{2} + \frac{1}{2}(D\xi +E\eta +2F) = 0.

Here $D\xi +E\eta +2F$ is the value of $\left(\frac{\partial f}{\partial z}\right)_{1}$ at $x=\xi$ , $y=\eta$ .

Problem 275

Translate the origin to the center of the curve $x^{2}+3xy-2y^{2}-7x-2y+16=0$ and set up the new equation of the curve.

Problem 276

Simplify the equation of the curve

x^{2}+3xy-2y^{2}-7x-2y+16=0,

by translating the origin to the center.

Problem 277

Simplify the equation of the curve

x^{2}-2xy+y^{2}-4x+4y-3=0,

by translating the origin to that center of it whose abscissa equals one.

Problem 278

Show that the line $7x+y+6=0$ passes through the centers of the curves

3x^{2}-7xy-6y^{2}+3x-9y+5=0

and

3x^{2}-5xy+6y^{2}+11x-17y+13=0.

Problem 279

Find the axes of symmetry of the ellipse

x^{2}+xy-y^{2}-x+y-1=0.

Problem 280

Find the foci (with respect to the old coordinate axes) of the hyperbola $x^{2}-6xy+y^{2}-2x-2y+5=0$ .

Problem 281

Determine the eccentricity of the curve

x^{2}-4xy-2y^{2}+10x+4y=0.

Problem 282

Simplify the equation of the curve

x^{2}-xy+y^{2}-5x+y-2=0,

by translating the origin to the center, and then rotating the axes through the appropriate angle.

Problem 283

The same, for the curve $x^{2}+6xy+y^{2}+8x+24y+39=0$ .

Problem 284

Simplify the equation of the parabola

x^{2}-2xy+y^{2}-2x-2y+1=0,

by rotating the axes so that the new equation contains no $x_{1}y_{1}$ , and then translating the origin so that only two terms remain in the final equation.

Problem 285

Prove that when $4AC-B^{2}=0$ , $A>0$ (so that one may set $A=a^{2}$ ), the identity holds

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F = (ax+by+h)^{2}+\alpha x+\beta y+\gamma ,

where $\alpha = D-2ah$ , $\beta = E-2bh$ , $\gamma = F-h^{2}$ .

Problem 286

Prove that in the identity of the preceding problem, the number $h$ can be chosen so that the lines $ax+by+h=0$ and $\alpha x+\beta y+\gamma =0$ are mutually perpendicular, provided only that the numbers $D$ and $E$ are not proportional to the numbers $a$ and $b$ .

Problem 287

Simplify the equation of the curve

x^{2}+2xy+y^{2}-9x+4=0

using the identity:

x^{2}+2xy+y^{2}-9x+4 = (x+y+h)^{2}+\alpha x+\beta y+\gamma ,

choosing $h$ so that the lines $\alpha x+\beta y+\gamma =0$ and $x+y+h=0$ are mutually perpendicular. Here the formulas for the coordinate transformation have the form:

y_{1} = \frac{x+y+h}{\sqrt{2}}, \qquad x_{1} = \frac{\alpha x+\beta y+\gamma }{\sqrt{\alpha ^{2}+\beta ^{2}}}.

Problem 288

By the same method, simplify the equation of the parabola

16x^{2}+24xy+9y^{2}-170x+310y-465=0.

Problem 289

Simplify the equation of the parabola

9x^{2}+30xy+25y^{2}+4x-16y+8=0.

Problem 290

For given $A$ , $B$ and $C$ , and a suitably chosen value of $\lambda$ , the identity holds

Ax^{2}+Bxy+Cy^{2}-\lambda (x^{2}+y^{2}) = (\alpha x+\beta y)^{2},

where $\alpha$ and $\beta$ are real or imaginary coefficients. Prove that such values of $\lambda$ are the roots of the equation

4\lambda ^{2}-4(A+C)\lambda +4AC-B^{2}=0.

Problem 291

The equation

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F = 0

represents a pair of lines, real or imaginary, if and only if the discriminant $\Delta$ equals zero. Using this, prove that under any coordinate transformation the quantity $\Delta$ does not change.

Remark.

The discriminant $\Delta$ of the quadratic form in three variables $f(x,y,z) = Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}$ is defined as the determinant of the coefficients of $x$ , $y$ and $z$ in the formulas expressing its partial derivatives: $\Delta = \begin{vmatrix} 2A & B & D \\ B & 2C & E \\ D & E & 2F \end{vmatrix}$ . Its minor $\delta = \begin{vmatrix} 2A & B \\ B & 2C \end{vmatrix} = 4AC-B^{2}$ is the discriminant of the quadratic form in two variables $Ax^{2}+Bxy+Cy^{2}$ .

Problem 292

Using the invariance of the quantities $A+C$ , $4AC-B^{2}$ and the discriminant $\Delta$ , prove that when $4AC-B^{2} \neq 0$ , the equation

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0

reduces, after simplification, to the form

\lambda _{1}x_{1}^{2}+\lambda _{2}x_{2}^{2} = \frac{\Delta }{B^{2}-4AC}.

Here $\lambda_{1}$ and $\lambda_{2}$ are the roots of the equation $\lambda^{2}-2(A+C)\lambda +4AC-B^{2}=0$ .

Problem 293

Prove that the equation $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ , when $AC-D^{2}=0$ and $\Delta \neq 0$ , can be reduced to the form

(A+C)x_{1}^{2} = \pm \sqrt{\frac{-\Delta }{2(A+C)}}y_{1}.

Problem 294

Reduce the equation of the following line to its simplest form: $5x^{2}+4xy+8y^{2}-32x-56y+80=0$ .

Problem 295

Reduce the equation of the following line to its simplest form: $5x^{2}+8xy+5y^{2}-18x-18y+9=0$ .

Problem 296

Reduce the equation of the following line to its simplest form: $5x^{2}+6xy+5y^{2}-16x-16y+16=0$ .

Problem 297

Reduce the equation of the following line to its simplest form: $5x^{2}-4xy+5y^{2}+4x-10y=319$ .

Problem 298

Reduce the equation of the following line to its simplest form: $6xy+8y^{2}-12x-26y+11=0$ .

Problem 299

Reduce the equation of the following line to its simplest form: $7x^{2}+16xy-23y^{2}-14x-16y-218=0$ .

Problem 300

Reduce the equation of the following line to its simplest form: $7x^{2}+24xy+38x+24y+175=0$ .

Problem 301

Reduce the equation of the following line to its simplest form: $x^{2}-8xy+7y^{2}+6x-6y+9=0$ .

Problem 302

Reduce the equation of the following line to its simplest form: $9x^{2}+24xy+16y^{2}-40x+30y=0$ .

Problem 303

Reduce the equation of the following line to its simplest form: $2x^{2}-72xy+23y^{2}+68x+26y+28=0$ .

Section 1.7Conjugate Diameters. Axes of Symmetry. Asymptotes[15/80%]

Problem 304

Find the equation of the hyperbola passing through the points $(1,1)$ , $(2,1)$ , $(-1,-2)$ and having the asymptote $x+y-1=0$ .

Problem 305

Find the equilateral hyperbola, given its asymptote $x-y+1=0$ and the points of the hyperbola $(1,1)$ , $(2,1)$ .

Problem 306

Set up the general equation of the equilateral hyperbolas with center at the point $(a,b)$ .

Problem 307

A parabola passes through the points $(0,0)$ and $(0,1)$ , and the line $x+y+1=0$ is its axis. Find the parabola.

Problem 308

The points $(0,1)$ , $(0,-1)$ , $(1,1)$ lie on a parabola. Its axis is parallel to the line $y=x$ . Find the parabola.

Problem 309

A parabola is tangent to the axis $Oy$ at the origin. The line $x+y+1=0$ is the tangent at the vertex. Find the parabola.

Problem 310

The line $x+y+1=0$ is the axis of a second-order curve, and the points $(0,0)$ , $(1,-1)$ , $(2,1)$ lie on the curve. Find the equation of the curve.

Problem 311

Prove that for the curve

A(ax+by+c)^{2}+B(\alpha x+\beta y+\gamma )^{2}+C=0

the lines $ax+by+c=0$ and $\alpha x+\beta y+\gamma =0$ are conjugate diameters, if $\alpha \beta \neq ab$ .

Problem 312

Prove that for the parabola

(ax+by+c)^{2}+A(\alpha x+\beta y+\gamma )=0

the line $\alpha x+\beta y+\gamma =0$ is a tangent, and the line $ax+by+c=0$ is the diameter conjugate to its direction. Here it is understood that $\alpha \beta \neq ab$ .

Problem 313

Find the second-order curve for which the lines $x-2y-1=0$ and $2x-y+1=0$ are conjugate diameters and on which the points $(1,0)$ and $(0,1)$ lie.

Problem 314

A parabola passes through the point $(0,0)$ ; the line $x+y-1=0$ is a diameter of the parabola, and $x+2y-1=0$ is the tangent conjugate to it. Find the parabola.

Problem 315

The line $x+y+1=0$ is a tangent, and $x-y+1=0$ is the conjugate diameter, of a parabola with parameter $\sqrt{2}$ . Find the parabola.

Problem 316

The lines $x+y-1=0$ and $x-y+1=0$ are conjugate diameters of an ellipse with semi-axes 2 and 1. Find the ellipse.

Problem 317

The lines $x+2y-4=0$ and $x-3y+2=0$ are conjugate diameters of an ellipse with semi-axes $\sqrt{2}$ and $\frac{\sqrt{3}}{3}$ . Find the equation of the ellipse.

Problem 318

Prove that the segments of any line contained between a hyperbola and its asymptotes are equal to one another.

Section 1.8Foci and Directrices[18/94%]

Problem 319

Find the focus and directrix of the parabola

x^{2}-2xy+y^{2}-6x-2y+9=0.

Problem 320

Find the foci and directrices of the curve

3x^{2}-4xy-2x+4y-5=0.

Problem 321

Find the foci and directrices of the curve

8x^{2}+4xy+5y^{2}+8x-16y-16=0.

Problem 322

A second-order curve passes through the origin, its focus is at the point $(-1,1)$ , and the corresponding directrix has the equation $x+y-2=0$ . Find the curve.

Problem 323

Find the ellipse passing through the point $(4,2)$ and having foci at the points $(4,3)$ and $(0,-1)$ .

Problem 324

Find the ellipse with foci at the points $(1,2)$ , $(2,1)$ , passing through the point $(5,5)$ .

Problem 325

Find the equilateral hyperbola from its directrix $x+y-1=0$ and focus $(1,1)$ .

Problem 326

Find the parabola from its points $(1,1)$ , $(1,2)$ and its directrix $x+y-1=0$ .

Problem 327

A second-order curve passes through the point $(0,-1)$ , has center $(1,1)$ , and has as directrix the line $x+2y+1=0$ . Find the curve.

Problem 328

Find the second-order curve from its points $(0,1)$ , $(1,0)$ , $(0,2)$ and focus $(1,1)$ .

Problem 329

Find the second-order curve from its directrix $x+y+1=0$ , axis $y=x$ , and points $(0,0)$ and $(1,0)$ .

Problem 330

Find the second-order curve from its points $(4,5)$ , $(-3,4)$ , focus $(1,1)$ , and axis of symmetry $x+y-2=0$ .

Problem 331

Find the parabola from its vertex $(0,0)$ and focus $(1,1)$ .

Problem 332

Find the hyperbola from its asymptotes $x+y-1=0$ , $x-y+1=0$ and focus $(0,2)$ .

Problem 333

Find the locus of the foci of parabolas passing through the point $(1,-1)$ and having as directrix the line $x+y+1=0$ .

Problem 334

Find the locus of the vertices of parabolas passing through a given point and having a given directrix.

Problem 335

Find the locus of the centers of equilateral hyperbolas passing through the point $(0,0)$ and having as directrix the line $x+y+1=0$ .

Problem 336

What is the locus of the foci of second-order curves inscribed in a given parallelogram?

Section 1.9Tangents to Second-Order Curves. Poles and Polars[13/54%]

Problem 337

Find the equation of the tangent at the origin to the curve

5x^{2}+7xy+y^{2}-x+2y=0.

Remark.

If the point $(x_{1},y_{1})$ lies on the second-order curve $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ , then the equation of the tangent to the curve at this point can be written in either of two forms: $(2Ax_{1}+By_{1}+D)x+(Bx_{1}+Cy_{1}+E)y+Dx_{1}+Ey_{1}+2F=0$ , $(2Ax+By+D)x_{1}+(Bx+Cy+E)y_{1}+Dx+Ey+2F=0$ . If, however, the point $(x_{1},y_{1})$ does not lie on the curve, the same equations represent a certain line, called the polar of the point $(x_{1},y_{1})$ . The point $(x_{1},y_{1})$ itself is called the pole of this polar.

Problem 338

Find the tangent to the curve $x^{2}-xy-y^{2}-2x+2y+1=0$ parallel to the line $2x+2y-1=0$ .

Problem 339

Find the equations of the tangents to the ellipse $x^{2}+xy+y^{2}=3$ parallel to the coordinate axes.

Problem 340

Find the equation of the tangent to the curve $3x^{2}+4xy+5y^{2}-7x-8y-3=0$ passing through the point $(2,1)$ .

Problem 341

Through the point $(4,-2)$ draw a tangent to the curve $x^{2}-xy-y^{2}-2x+2y+1=0$ .

Problem 342

Prove the theorem: if a pole moves along a line, its polar rotates about a certain point.

Problem 343

Prove the theorem: if the polar of a point $M$ passes through a point $N$ , then the polar of the point $N$ passes through the point $M$ .

Problem 344

Prove Brianchon's theorem: the diagonals joining opposite vertices of a hexagon circumscribed about a second-order curve meet at one point.

Problem 345

Prove that a second-order curve is determined by five of its tangents.

Problem 346

Prove that the poles of the tangents to a certain second-order curve, taken with respect to a given second-order curve, lie on a second-order curve.

Problem 347

Find the locus of the poles of the tangents to a circle of radius $R$ with respect to a concentric circle of radius $r$ .

Problem 348

If one of the sides vanishes, or two sides are extensions of one another, the hexagon becomes a pentagon. What do Pascal's theorem (problem 270) and Brianchon's theorem (problem 344) become in this case?

Problem 349

The same question, for the further degenerate cases in which the hexagon becomes a quadrilateral or a triangle.

Section 1.10Miscellaneous Problems[34/53%]

Problem 350

Prove that the area contained between a tangent to a hyperbola and its asymptotes is a constant quantity.

Problem 351

Prove that the product of the segments of a secant between the asymptotes and a point on the hyperbola equals the square of the semi-diameter parallel to the secant.

Problem 352

Prove that for a hyperbola, the point of tangency bisects the segment of the tangent between the asymptotes.

Problem 353

Find the equation of the second-order curve from its focus $(0,0)$ and three tangents: $x-y-1=0$ , $2x-y-1=0$ , $x+y-1=0$ .

Problem 354

Find the parabola from a point on it $(5,0)$ , focus $(3,2)$ , and tangent $x-3y-7=0$ .

Problem 355

Find the hyperbola from its center $(1,1)$ , focus $(3,3)$ , and tangent $x+2y-7=0$ .

Problem 356

Find the locus of the points of tangency of the tangents drawn from the origin to the parabola

\left((x-a)^{2}+y^{2}-b^{2}\right)(1+m^{2})-(y-mx)^{2}=0,

where $m$ is a variable parameter.

Problem 357

Find the locus of the projections of the focus onto the tangents to the curve $\frac{x^{2}}{a^{2}}\pm \frac{y^{2}}{b^{2}}=1$ .

Problem 358

Find the point of intersection of the polars of points lying on the directrix.

Problem 359

If $a$ and $b$ are the lengths of two mutually perpendicular tangents to the parabola $y^{2}=4x$ , then $a^{4}b^{4}=(a^{2}+b^{2})^{3}$ . Prove this.

Problem 360

Find the locus of the points of tangency, for tangents of a given direction, to the confocal curves $\frac{x^{2}}{a^{2}+h}+\frac{y^{2}}{b^{2}+h}=1$ , where $h$ is a variable parameter.

Problem 361

An equilateral hyperbola passes through the origin and has the asymptote $x+y+1=0$ . Find the locus of the points of intersection of the second asymptote with the tangent at the origin.

Problem 362

Equilateral hyperbolas pass through the point $(1,0)$ and are tangent to the axis $Oy$ at the point $(0,1)$ . Find the locus of their centers.

Problem 363

What is the locus of the vertices of equilateral hyperbolas passing through a given point and having a given asymptote?

Problem 364

What is the locus of points from which two normals at right angles to one another can be drawn to a given parabola?

Problem 365

The parabolas $y^{2}=2px$ and $y^{2}=2p(x-a)$ are given. Prove that the chords of the first that are tangent to the second are bisected by the point of tangency.

Problem 366

Two parabolas have a common axis of symmetry. Two lines are parallel to the axis of symmetry. Chords are drawn through the points of their intersection with the parabolas. Prove that the points of intersection of these chords lie on one line, whichever pair of parallel lines is taken.

Problem 367

What is the locus of the vertices of parabolas with a given focus and tangent?

Problem 368

Prove that the circle circumscribed about the triangle formed by three tangents to a parabola passes through the focus.

Problem 369

Prove that the equilateral hyperbola circumscribed about a triangle passes through the point of intersection of its altitudes.

Problem 370

What is the locus of the foci of parabolas with a given vertex and tangent?

Problem 371

From a point outside a parabola, three normals can be drawn to it. Prove that the vertex of the parabola and the three feet of the normals lie on one circle.

Problem 372

The curve $Ax^{2}+By^{2}+C=0$ and the point $M(a,b)$ are given. About $M$ as center, circles are described. Find the locus of the midpoints of the chords common to these circles and the given curve.

Problem 373

A quadrilateral $ABCD$ is given. What is the locus of points $M$ such that the sum of the areas of the triangles $AMB$ and $CMD$ equals the sum of the areas of the triangles $BMC$ and $DMA$ ?

Problem 374

Using the result of the preceding problem, prove Newton's theorem: in a quadrilateral circumscribed about a circle, the midpoints of the diagonals and the center of this circle lie on one line.

Problem 375

A right angle, whose vertex $M$ lies on a second-order curve, rotates about the vertex, and $A$ and $B$ are the points of intersection of its sides with the curve, other than $M$ . Prove that the line $AB$ rotates about a point lying on the normal to the curve at the point $M$ .

Problem 376

A right angle rotates about a point $A$ inside a given circle. At the points of intersection of its sides with the circle, tangents to the circle are drawn. Find the locus of the points of intersection of these tangents.

Problem 377

A variable circle is tangent to an ellipse at a given point. What is the locus of the points of intersection of the common tangents?

Problem 378

The foci of an ellipse are joined by radius vectors to a variable point of the ellipse. What is the locus of the centers of the circles inscribed in the triangles formed by the focal chord and the radius vectors?

Problem 379

Inversion, or transformation by reciprocal radius vectors, consists in replacing a variable point $M$ by another point $M_{1}$ , lying on the line $OM$ , such that $OM \cdot OM_{1} = a^{2}$ . Here $O$ is a given point, and $a$ is a given quantity. Prove that under inversion the line $A(x^{2}+y^{2})+Bx+Cy+D=0$ passes into the line $A_{1}(x^{2}+y^{2})+B_{1}x+C_{1}y+D_{1}=0$ , i.e., circles or lines pass into circles or lines (not necessarily respectively).

Problem 380

Prove that by inversion with respect to the origin, the hyperbola $x^{2}-y^{2}=a^{2}$ is carried into the lemniscate of Bernoulli:

(x^{2}+y^{2})^{2} = b^{2}(x^{2}-y^{2}).

Problem 381

Prove that the inversion of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with respect to the origin, with radius of inversion equal to $\sqrt{ab}$ , gives the curve $(x^{2}+y^{2})^{2}=b^{2}x^{2}+a^{2}y^{2}$ .

Problem 382

A hyperbola congruent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ , where $a>b$ , is inscribed in the angle between the coordinate axes and slides along them. Prove that its center moves along the circle $x^{2}+y^{2}=a^{2}-b^{2}$ .

Problem 383

An ellipse with semi-axes $a$ and $b$ is tangent to the coordinate axes and slides along them. Prove that its center moves along the circle $x^{2}+y^{2}=a^{2}+b^{2}$ .