Chapter III

Reduce to canonical form the equation

Reduce to canonical form the equation

What class of curve can a second-order equation define if its left-hand side factors into a product of linear polynomials?

Under what necessary and sufficient condition on its coefficients is a second-order equation in a Cartesian rectangular coordinate system the equation of a circle?

A coordinate system satisfies the conditions $\left|\mathbf{e}_{1}\right|=\left|\mathbf{e}_{2}\right|=5,\left(\mathbf{e}_{1}, \mathbf{e}_{2}\right)=7$. What curve is defined in this coordinate system by the equation $x^{2}+y^{2}=1$ ?

Prove that the sum of the coefficients $A+C$ in the general second-order equation does not change under the passage from one Cartesian rectangular coordinate system to another such system.

Prove that the vertices of a hyperbola and the points of intersection of its asymptotes with the directrices lie on one circle.

The focus of an ellipse (hyperbola or parabola) divides a chord passing through it into segments of lengths $u$ and $v$. Prove that the sum $1 / u+1 / v$ is constant.

Derive the equations of the ellipse, the hyperbola and the parabola in a polar coordinate system, taking as the pole a focus and as the polar axis the ray lying on the axis of symmetry and not intersecting the directrix corresponding to that focus.

An ellipse and a parabola are drawn in the plane together with their axes of symmetry. How can one construct their foci and directrices with compass and straightedge? The same question for a hyperbola whose asymptotes are drawn. (The problem of constructing the axes of symmetry and the asymptotes is solved on the basis of the material of § 3.)

Let $u$ and $v$ be the lengths of two mutually perpendicular radii of an ellipse. Find the sum $1 / u^{2}+1 / v^{2}$.

Find the shortest distance from the parabola $y^{2}=12 x$ to the line $x-y+7=0$.

Prove that the segment of a tangent enclosed between the asymptotes of a hyperbola is bisected by the point of tangency.

In the equation of the tangent to an ellipse in the canonical coordinate system, instead of the coordinates $x_{0}$ and $y_{0}$ of the point of tangency there are substituted the coordinates of a point lying not on the ellipse but outside the ellipse. How is the resulting line situated?

From a point on the directrix two tangents to a parabola are drawn. Prove that they are mutually perpendicular and that the segment joining the points of tangency passes through the focus.

A curve is said to be circumscribed about a parallelogram if its vertices lie on the curve and the remaining points do not lie on it. Prove that such a second-order curve is necessarily central, and that its center coincides with the center of the parallelogram.

An ellipse, a hyperbola and a parabola are drawn in the plane. How can one construct with compass and straightedge their axes of symmetry and the asymptotes of the hyperbola?

Prove that the sum of the squares of the lengths of the chords lying on conjugate diameters of an ellipse is constant.

Without reducing the equation to canonical form, find the center and the asymptotes of the hyperbola

Without reducing the equation to canonical form, say what the curve is called whose equation is

How can one factor the left-hand side of the equation from problem 5?

Write the equation of the tangent to the curve $x^{2}-2 x y+3 y^{2}=3$ at the point $M_{0}(0,1)$.

Determine to what class of second-order curves the curve with the equation

in a Cartesian coordinate system belongs.

Prove that the curve of intersection of a second-order surface with a plane that does not lie entirely on it is an algebraic curve of order at most two. How many common points can a line and a second-order surface have?

Find the equation and determine the type of the surface obtained by rotating about the axis of applicates the straight line:

Prove that the rectilinear generators of a hyperbolic paraboloid belonging to one family are all parallel to some plane.

On the hyperbolic paraboloid with the equation

there lie the parabolas $y=0, x^{2}=2 a^{2} z$ and $x=0, y^{2}=-2 b^{2} z$. Let the points $A_{1}$ and $B_{1}$ on the first parabola and the points $A_{2}$ and $B_{2}$ on the second all be at the same distance from the plane $z=0$.

Prove that the lines $A_{1} B_{2}, A_{1} A_{2}, B_{1} A_{2}$ and $B_{1} B_{2}$ are rectilinear generators.

Find the projection onto the plane $z=0$ of the curve of intersection of the two-sheeted hyperboloid $-x^{2}+y^{2}-z^{2}=1$ and the cone $5 x^{2}-3 y^{2}+4 z^{2}=0$.

Prove that no plane intersects an elliptic paraboloid in a hyperbola.