The Line and the Circle

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

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

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

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

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 $y = 2x-3$.

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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 $\left|OM'\right| : \left|OM\right|$.

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

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.

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

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

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.

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

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

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

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 $1:3$.

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

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.

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 $M$.

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.

$M_{1}(2,1)$ 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.

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

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

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

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

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.

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

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

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

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.

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 $x-y-1=0$.

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

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 $3\sqrt{2}$.

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

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

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 $(-1,-1)$.

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

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

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.

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

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 $C$.

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.

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.

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.

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.

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.

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.

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 $AB$.

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

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.

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.

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

(Menelaus's theorem).

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

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.

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.

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.

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

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 $Ox$.

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

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

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

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

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

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

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

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

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

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

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.

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 $x_{1}^{2}+y_{1}^{2}+Ax_{1}+By_{1}+C$.

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

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

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.

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

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.)

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.)

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.

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.

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.