Second-Order Curves

In each case, set up the canonical equation of an ellipse:

In each of the following cases, set up the canonical equation of a hyperbola:

A circle passes through the points $(1,4),(-7,4)$ and $(2,-5)$. Find its center, radius, and write its equation.

Write the equation of an ellipse intersecting the axis $O x$ at the points $(1,0)$ and $(9,0)$ and tangent to the axis $O y$ at the point $(0,3)$, knowing that its axes are parallel to the coordinate axes.

Write the equation of an ellipse with vertices $(0,6)$ and $(0,-2)$, knowing that on the axis $O x$ this ellipse cuts off a chord of length 6.

Write the equation of a rectangular (equilateral) hyperbola, one of whose vertices is at the point $(2,2)$, whose real axis is parallel to the axis $O y$, and which cuts off a chord of length 8 on the axis $O x$.

Set up the equation of a parabola, if the coordinates of the focus $F(3,0)$ and the equation of the directrix $x = -1$ are given.

Write the equation of a parabola whose vertex is at the point $(2,6)$, and whose axis is parallel to the axis $O y$, knowing that on the axis $O x$ this parabola cuts off a chord of length 6.

Write the equations of the ellipse and hyperbola with foci $(7,0)$ and $(-7,0)$, passing through the point $(-2,12)$.

Set up the equation of an ellipse whose foci have coordinates $(0,1)$ and $(1,0)$, and whose major axis equals 2.

The coordinates of the foci $F_{1}(1,0)$ and $F_{2}(0,1)$ and the equation of a tangent line $x+y-2 = 0$ are given. Set up the equation of the ellipse.

Write the equation of a parabola, knowing its focus $F(-1,-2)$ and directrix $x-y+8 = 0$.

Write the equation of a parabola whose vertex is at the origin, and whose focus is at the point $F(1,1)$.

Write the equation of a second-order curve, knowing its focus $F(0,1)$, the directrix $x-y+3 = 0$ corresponding to the focus $F$, and a point $A(7,0)$ lying on it.

Set up the equation of a second-order curve, knowing one of its foci $F(1,1)$, the directrix corresponding to it, $x+2 y-1 = 0$, and the eccentricity $e = \sqrt{5}$.

Write the equation of a second-order curve, knowing its foci $F_{1}(1,1)$ and $F_{2}(-2,-2)$ and one of its directrices, $x+y-1 = 0$.

Write the equation of a second-order curve with center at the point $C(1,2)$, passing through the origin, one of whose directrices is the line $x+2 y-1 = 0$.

Write the equation of a rectangular (equilateral) hyperbola for which the axis $O x$ is an asymptote, and the point $(1,1)$ is a vertex.

The equations of the asymptotes of a hyperbola, $y = \pm 5 x / 12$, and a point on it, $M(24,5)$, are given. Set up the canonical equation of the hyperbola.

The equations $A_{1} x+B_{1} y+C_{1} = 0$ and $A_{2} x+B_{2} y+C_{2} = 0$ of the asymptotes of a hyperbola, and the coordinates of a point $M\left(x_{0}, y_{0}\right)$ lying on the hyperbola, are given. Set up the equation of the hyperbola.

Find the locus of the midpoints of the segments cutting off, from a given angle of magnitude $\alpha$, triangles of fixed area $S$. Prove that all these segments are tangent to one and the same hyperbola.

Prove that the product of the distances from any point of a hyperbola to its two asymptotes is the same for all points of the hyperbola.

Write the equation of a hyperbola for which the point $F(-2,2)$ is a focus, and the lines $2 x-y+1 = 0, x+2 y-7 = 0$ are asymptotes.

Write the equation of a hyperbola, knowing the asymptote $y = 0$, one of its axes $2 x-y+2 = 0$, and a point $(1,1)$ lying on the hyperbola.

Write the equation of a parabola that passes through the points $O(0,0), P(4,0), Q(0,2)$, under the condition that the points $P$ and $Q$ are symmetric with respect to the axis of this parabola.

Write the equation of an ellipse with semi-axis lengths $a = 2, b = 1$, for which the lines $x+y-1 = 0$ and $x-y+1 = 0$ are, respectively, the major and minor axes.

Write the equation of a second-order curve passing through the points $M_{1}(-2,-1)$ and $M_{2}(0,-2)$, for which the lines $x+y+1 = 0$ and $x-y+1 = 0$ are axes of symmetry.

Write the equation of a parabola that passes through the points $(0,0),(0,1)$, for which the line $x+y+1 = 0$ is an axis of symmetry.

Write the equation of an ellipse whose center is at the point $(4,3)$, one vertex lies at the origin, and the other, adjacent to it, on the axis $O y$.

Set up the equation of a hyperbola, if the coordinates of two of its foci, $F_{1}\left(x_{1}, y_{1}\right)$ and $F_{2}\left(x_{2}, y_{2}\right)$, and the equation of a tangent line, $A x+B y + C = 0$, are given.

Write the equation of a second-order curve for which the axis $O x$ is an axis of symmetry, the axis $O y$ is the tangent at the vertex, knowing that the curve passes through the two points $(2,3)$ and $(6,-3)$.

Find the foci and directrices of the rectangular (equilateral) hyperbola $2 x y = a^{2}$.

Find the foci of the hyperbola:

Find the coordinates of the focus of the parabola $x^{2} = 4 y$.

Set up the equation of the directrix of the parabola $y^{2} = 6 x$.

Find the foci and corresponding directrices of the following curves:

Using a translation of the coordinate axes, find the canonical equation and determine the position of the second-order curves given by the equations:

Find the centers of the following second-order curves:

Find the canonical form and the formulas relating the original coordinates to the canonical ones, for the following second-order curves:

Determine the canonical form and canonical coordinate system of the following second-order curves:

In a rectangular coordinate system, a second-order curve is given by the equation $6 x y+8 y^{2}-12 x-26 y+11 = 0$.

Set up the canonical equations of the curves, if their invariants are given:

Prove that the conditions $I_{1}^{2} = 4 I_{2}, I_{1} I_{3} < 0$ are necessary and sufficient for the equation of a second-order curve to define a real circle.

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$, the conditions necessary and sufficient for the second-order equation $F(x, y) = 0$ to define a real ellipse. Express, in terms of $I_{1}, I_{2}, I_{3}$, the lengths $a$ and $b$ of the semi-axes of the ellipse, and find its area $S$.

Using invariants, express the necessary and sufficient conditions for the equation $F(x, y) = 0$ to define a hyperbola lying in the acute angle formed by its asymptotes.

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$, the conditions necessary and sufficient for the quadric $F(x, y) = 0$ to split into a pair of perpendicular lines.

Prove that the equation $F(x, y) = 0$ defines a rectangular (equilateral) hyperbola if and only if $I_{1} = 0$ and $I_{3} \neq 0$.

The equation $F(x, y) = 0$ defines a hyperbola. Using orthogonal invariants, write the equation of the pair of its asymptotes.

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$, the necessary and sufficient conditions for the curve $F(x, y) = 0$ to be a parabola. Express, in terms of $I_{1}, I_{2}, I_{3}$, its focal parameter $p$.

The equation $F(x, y) = 0$ defines a pair of intersecting lines. Express, in terms of orthogonal invariants, the angle between them.

(Semi-invariant.) Prove that the quantity

is an invariant of equation (6.1) under homogeneous orthogonal transformations (i.e., those under which the origin does not change).

Prove that if $I_{2} = I_{3} = 0$, then $I_{2}^{*}$ is preserved under any orthogonal transformations.

A second-order equation $F(x, y) = 0$ defines two parallel lines. Express the distance $d$ between them in terms of the invariants and the semi-invariant of the polynomial $F(x, y)$.

A second-order curve, given by the equation $F(x, y) = 0$, splits into a pair of intersecting, not mutually perpendicular lines. Find the necessary and sufficient condition for a given point $M_{0}\left(x_{0}, y_{0}\right)$ to lie in the acute angle formed by these lines.

A second-order curve, given by the equation $F(x, y) = 0$, splits into a pair of parallel lines. Under what necessary and sufficient condition does a given point $M_{0}\left(x_{0}, y_{0}\right)$ lie between them?

The equation $a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2} = 0$ defines a pair of intersecting lines $(a_{11} a_{22}-a_{12}^{2} < 0)$. Prove that if the coordinate system is rectangular, then the pair of bisectors between these lines can be given by the equation

The equations of two hyperbolas have the form

Under what necessary and sufficient condition will they lie in different vertical angles formed by their common asymptotes?

Determine the affine type of the following curves:

Using Lagrange's method, determine the affine type of the following second-order curves:

Find the necessary and sufficient condition for the line $A x+B y+C = 0$ to:

Set up the equation of the tangent to the quadric

at the origin.

Set up the equations of the tangents to the ellipse

drawn from the point $(12,-3)$.

Set up the equations of the tangents to the ellipse

passing through the point:

Set up the equation of the tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{36} = 1$:

Set up the equations of the tangents to the hyperbola

passing through the point:

Set up the equation of the tangents to the parabola $y^{2} = 16 x$, passing through the point:

Draw tangents to the quadric $2 x^{2}+7 x y+6 y^{2}+3 x + 5 y-26 = 0$ through the points:

Find the conditions necessary and sufficient for it to be possible to draw from the point $\left(x_{0}, y_{0}\right)$ to the hyperbola:

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ and the ellipse

Prove that for any value of $\theta$ the line

is tangent to one and the same ellipse. Set up its equation.

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ with the hyperbola

Prove that for any value of $\theta$ the line

is tangent to one and the same hyperbola, given by its canonical equation. Set up the equation of this hyperbola.

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ and the parabola $y^{2} = 2 p x$.

(Dual quadric.) Prove that the line $\alpha x+\beta y + 1 = 0$ is tangent to the nondegenerate real quadric (6.1) or is one of its asymptotes if and only if $\alpha$ and $\beta$ satisfy the equation

Let $\Gamma$ be a second-order curve for which $I_{2} \neq 0$ or $I_{3} \neq 0$. Prove that the line $A x+B y+C = 0$ is tangent to the curve $\Gamma$, passes through its singular point, or is an asymptote if and only if the following equality holds:

Prove that for any value of $k$ the line $y = k x+1/k$ is tangent to one and the same parabola, given by its canonical equation. Find the equation of this parabola.

Set up the equation of the common tangents to the ellipses

An ellipse having foci at the points $F_{1}(-3,0)$ and $F_{2}(3,0)$ is tangent to the line $x+y-5 = 0$. Set up the equation of the ellipse.

Set up the equation of a second-order curve that is tangent to the lines $x = 0, y = 0$ at the points $P(0,3), Q(4,0)$ and to the line $x+y-1 = 0$. Find the point of tangency with the last line.

Write the equation of a hyperbola having as asymptotes the lines $x-1 = 0,2 x-y+1 = 0$ and tangent to the line $4 x+y+5 = 0$.

Set up the equation of a parabola tangent to the axes $O x$ and $O y$ at the points $P(3,0)$ and $Q(0,5)$, respectively.

Set up the equation of a second-order curve tangent to the sides of a triangle $P Q R$ and having its center at the point $S(2,1)$, if the coordinates of the vertices $P(0,0), Q(5,0), R(0,4)$ are known.

Prove that the product of the distances from the foci of an ellipse to any tangent to it equals the square of the minor semi-axis.

Prove that the product of the distances from the foci of a hyperbola to any tangent to it equals the square of the imaginary semi-axis.

The foci of a hyperbola, $F_{1}(4,2), F_{2}(-1,-10)$, and the equation of a tangent line, $3 x+4 y-5 = 0$, are given. Find the semi-axes of the hyperbola.

Find the locus of the points of intersection of mutually perpendicular tangents to a given ellipse with semi-axes $a, b$.

An ellipse with semi-axes $a$ and $b$, where $a < b$, moving in the plane remains tangent to the coordinate axes while staying in the first quadrant. What curve does the center of the ellipse trace?

Find the locus of the vertices of right angles whose sides are tangent to a given hyperbola with semi-axes $a$ and $b$.

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

Set up the equation of the line passing through the midpoints of the chords $2 x-y+7 = 0,2 x-y-1 = 0$ of the ellipse

Set up the equation of the diameter of the parabola $y = x^{2} / 6$, conjugate to the chord $y = 1.5 x+4$.

Write the equation of the diameter of the second-order curve

passing through the midpoint of the chord cut off by this curve on the line $x-2 y-1 = 0$.

Find the chord of the parabola $y^{2} = 4 x$ that is bisected by the point $(3,1)$.

Set up the equation of the chord of the ellipse

that is bisected by the point $M(2,1)$.

For the curve $x^{2}+2 x y-3 y^{2}+4 x+5 y-6 = 0$, draw the chord that is bisected by the point:

Find the common diameter of the two curves:

What set does equation (6.3) define, if a vector of asymptotic direction is taken as $(\alpha , \beta )$, for:

Find the asymptotes of the hyperbola

Two second-order curves are given

For each curve, find a pair of conjugate diameters such that the diameters of one pair are parallel to the diameters of the other pair.

Find the axes of symmetry of the following second-order curves (the coordinate system is rectangular):

In the equation $F(x, y) = 0$ of a second-order curve, the following change of coordinates is made: $x = x^{\prime }+x_{0}, y = y^{\prime }+y_{0}$, where $(x_{0}, y_{0})$ is a certain point. Prove that in the resulting equation:

Two noncollinear vectors $\mathbf{e}_{1} = \left(\alpha_{1}, \beta_{1}\right)$ and $\mathbf{e}_{2} = \left(\alpha_{2}, \beta_{2}\right)$, given relative to an affine coordinate system $O x y$, are taken as the basis of a new coordinate system $O x^{\prime } y^{\prime }$. Suppose that in the new system the curve (6.1) is given by the equation

Prove that the coefficients of the new equation are expressed in terms of the coefficients of the old one as follows:

Write the equation of an ellipse, taking as the origin of an affine coordinate system its center, and as basis points $(1,0)$ and $(0,1)$ the endpoints of two mutually conjugate diameters.

Write the equation of an ellipse, knowing its center $C(2,1)$ and the endpoints of two conjugate diameters, $A(5,1), B(0,3)$.

Write the equation of an ellipse, knowing that its center is at the point $C(2,1)$ and that the lines $y-2 = 0$ and $x-y = 0$ are tangents at the endpoints of two conjugate diameters.

How is the equation of a parabola written, if an arbitrary point of the parabola is chosen as the origin of an affine coordinate system, the tangent at that point as the axis $O y$, the diameter conjugate to this tangent as the axis $O x$, and some point of the parabola as the unit point $E$?

The equation