Conjugate Diameters. Axes of Symmetry. Asymptotes

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

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

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

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

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

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.

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.

Prove that for the curve

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

Prove that for the parabola

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 $\alpha \beta \neq ab$.

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.

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.

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.

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.

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.

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