§ 3

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.