If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2 = 2x - 3$ intersect at the point $R(0, 1)$, then the orthocentre of the triangle $PQR$ is
(1) $(0, 1)$
(2) $(2, -1)$
(3) $(6, 3)$
(4) $(2, 1)$

Let the focal chord of the parabola $P : y ^ { 2 } = 4 x$ along the line $L : y = m x + c , m > 0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $O M F N$ is
(1) $2 \sqrt { 6 }$
(2) $2 \sqrt { 14 }$
(3) $4 \sqrt { 6 }$
(4) $4 \sqrt { 14 }$

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9} - \frac{y^2}{4} = 1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is
(1) 18
(2) 26
(3) 22
(4) 20

Let $\mathrm { P } ( 4,4 \sqrt { 3 } )$ be a point on the parabola $y ^ { 2 } = 4 \mathrm { a } x$ and PQ be a focal chord of the parabola. If M and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
(1) $17 \sqrt { 3 }$
(2) $\frac { 263 \sqrt { 3 } } { 8 }$
(3) $\frac { 34 \sqrt { 3 } } { 3 }$
(4) $\frac { 343 \sqrt { 3 } } { 8 }$

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y ^ { 2 } = 4 x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to y-axis. If the diagonal AC is of length $\frac { 25 } { 4 }$ and it passes through the point $( 1,0 )$, then the area of $ABCD$ is
(1) $\frac { 75 } { 4 }$
(2) $\frac { 25 } { 2 }$
(3) $\frac { 125 } { 8 }$
(4) $\frac { 75 } { 8 }$

Let A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If d denotes the distance between A and B, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y ^ { 2 } = 4 x$, then the value of $( a + d )$ is

Let the ellipse $\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2} + \frac{y^2}{\mathrm{b}^2} = 1,\ \mathrm{a} > \mathrm{b}$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{A}^2} + \frac{y^2}{\mathrm{B}^2} = 1,\ \mathrm{A} < \mathrm{B}$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $ABCD$ equals:
(1) $\frac{12\sqrt{6}}{5}$
(2) $6\sqrt{6}$
(3) $\frac{18\sqrt{6}}{5}$
(4) $\frac{24\sqrt{6}}{5}$