Deriving Equations of 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 $(6,-3)$.