For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$ make a perspective drawing illustrating what precedes.

Consider the branch of the rectangular hyperbola $xy = 1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) None of the above.

Let a line with slope of $60 ^ { \circ }$ be drawn through the focus $F$ of the parabola $y ^ { 2 } = 8 ( x + 2 )$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$-axis at the point $P$, then the length of the segment PF is
(a) $16 / 3$
(b) $8 / 3$
(c) $16 \sqrt{3} / 3$
(d) $8 \sqrt{3}$

From a point $P(h,k)$, two tangents are drawn to the parabola $y^2 = 4ax$ which are perpendicular to each other. Find the locus of $P$.

Let $P$ be a point on the ellipse $x^2 + 4y^2 = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $PC : PD$ equals
(A) 2
(B) $1/2$
(C) 4
(D) $1/4$

Let $P$ be a point on the ellipse $x^2 + 4y^2 = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $PC : PD$ equals
(A) 2
(B) $1/2$
(C) 4
(D) $1/4$

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(A) 2
(B) $1 / 2$
(C) 4
(D) $1 / 4$

Let $a , b , c , d > 0$, be any real numbers. Then the maximum possible value of $c x + d y$, over all points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, must be
(A) $\sqrt { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } }$.
(B) $\sqrt { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } }$.
(C) $\sqrt { \frac { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } } { a ^ { 2 } + b ^ { 2 } } }$.
(D) $\sqrt { \frac { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } } { c ^ { 2 } + d ^ { 2 } } }$.

Let $y = x + c _ { 1 } , y = x + c _ { 2 }$ be the two tangents to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 1$. What is the value of $\left| c _ { 1 } - c _ { 2 } \right|$?
(A) $\sqrt { 2 }$
(B) $\sqrt { 5 }$
(C) $\frac { \sqrt { 5 } } { 2 }$
(D) 1

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(a) 2 .
(B) $1 / 2$.
(C) 4 .
(D) $1 / 4$.

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2 + 4y^2 = 12$. Then its equation is
(A) $x^2\csc^2\theta - y^2\sec^2\theta = 1$
(B) $x^2\sec^2\theta - y^2\csc^2\theta = 1$
(C) $x^2\sin^2\theta - y^2\cos^2\theta = 1$
(D) $x^2\cos^2\theta - y^2\sin^2\theta = 1$

Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$ is
(A) $1 - \sqrt { \frac { 2 } { 3 } }$
(B) $\sqrt { \frac { 3 } { 2 } } - 1$
(C) $1 + \sqrt { \frac { 2 } { 3 } }$
(D) $\sqrt { \frac { 3 } { 2 } } + 1$

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right) , y _ { 1 } < 0 , y _ { 2 } < 0$, be the end points of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. The equations of parabolas with latus rectum $P Q$ are
(A) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(B) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(C) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$
(D) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$

The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A , M$ and the origin $O$ is
(A) $\frac { 31 } { 10 }$
(B) $\frac { 29 } { 10 }$
(C) $\frac { 21 } { 10 }$
(D) $\frac { 27 } { 10 }$

An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
(A) Equation of ellipse is $x^{2}+2y^{2}=2$
(B) The foci of ellipse are $(\pm1,0)$
(C) Equation of ellipse is $x^{2}+2y^{2}=4$
(D) The foci of ellipse are $(\pm\sqrt{2},0)$

The tangent $PT$ and the normal $PN$ to the parabola $y^{2}=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
(A) vertex is $\left(\frac{2a}{3},0\right)$
(B) directrix is $x=0$
(C) latus rectum is $\frac{2a}{3}$
(D) focus is $(a,0)$

Match the conics in Column I with the statements/expressions in Column II.
Column I
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Column II
(p) The locus of the point $( h , k )$ for which the line $h x + k y = 1$ touches the circle $x ^ { 2 } + y ^ { 2 } = 4$
(q) Points $z$ in the complex plane satisfying $| z + 2 | - | z - 2 | = \pm 3$
(r) Points of the conic have parametric representation $x = \sqrt { 3 } \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } \right) , y = \frac { 2 t } { 1 + t ^ { 2 } }$
(s) The eccentricity of the conic lies in the interval $1 \leq x < \infty$
(t) Points $z$ in the complex plane satisfying $\operatorname { Re } ( z + 1 ) ^ { 2 } = | z | ^ { 2 } + 1$

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The coordinates of $A$ and $B$ are
A) $( 3,0 )$ and $( 0,2 )$
B) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$
C) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $( 0,2 )$
D) $(3, 0)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$

Let A and B be two distinct points on the parabola $\mathrm { y } ^ { 2 } = 4 \mathrm { x }$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining A and B can be
A) $- \frac { 1 } { r }$
B) $\frac { 1 } { r }$
C) $\frac { 2 } { r }$
D) $- \frac { 2 } { \mathrm { r } }$

The line $2 \mathrm { x } + \mathrm { y } = 1$ is tangent to the hyperbola $\frac { \mathrm { x } ^ { 2 } } { \mathrm { a } ^ { 2 } } - \frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

Let $P ( 6,3 )$ be a point on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal at the point $P$ intersects the $x$-axis at $( 9,0 )$, then the eccentricity of the hyperbola is
(A) $\sqrt { \frac { 5 } { 2 } }$
(B) $\sqrt { \frac { 3 } { 2 } }$
(C) $\sqrt { 2 }$
(D) $\sqrt { 3 }$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
Length of chord $P Q$ is
(A) $7 a$
(B) $5 a$
(C) $2 a$
(D) $3 a$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
If chord $P Q$ subtends an angle $\theta$ at the vertex of $y ^ { 2 } = 4 a x$, then $\tan \theta =$
(A) $\frac { 2 } { 3 } \sqrt { 7 }$
(B) $\frac { - 2 } { 3 } \sqrt { 7 }$
(C) $\frac { 2 } { 3 } \sqrt { 5 }$
(D) $\frac { - 2 } { 3 } \sqrt { 5 }$

A vertical line passing through the point $( h , 0 )$ intersects the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta ( h ) =$ area of the triangle $P Q R$, $\Delta _ { 1 } = \max _ { 1/2 \leq h \leq 1 } \Delta ( h )$ and $\Delta _ { 2 } = \min _ { 1/2 \leq h \leq 1 } \Delta ( h )$, then $\frac { 8 } { \sqrt { 5 } } \Delta _ { 1 } - 8 \Delta _ { 2 } =$