Let Q be the point where the tangent line at point $\mathrm { P } ( a , b )$ on the parabola $y ^ { 2 } = 4 x$ meets the $x$-axis. When $\overline { \mathrm { PQ } } = 4 \sqrt { 5 }$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) 21
(2) 32
(3) 45
(4) 60
(5) 77

Let $d$ be the distance between the focus of the parabola $y ^ { 2 } = n x$ and the tangent line to the parabola at the point $( n , n )$. Find the minimum natural number $n$ satisfying $d ^ { 2 } \geq 40$. [4 points]

The tangent line at the point $( b , 1 )$ on the hyperbola $x ^ { 2 } - 4 y ^ { 2 } = a$ is perpendicular to one asymptote of the hyperbola. What is the value of $a + b$? (Given that $a , b$ are positive numbers.) [3 points]
(1) 68
(2) 77
(3) 86
(4) 95
(5) 104

For a natural number $n$ ($n \geq 2$), let the line $x = \frac{1}{n}$ meet the two ellipses $$C_{1} : \frac{x^{2}}{2} + y^{2} = 1, \quad C_{2} : 2x^{2} + \frac{y^{2}}{2} = 1$$ at points P and Q respectively in the first quadrant. Let $\alpha$ be the $x$-intercept of the tangent line to ellipse $C_{1}$ at point P, and let $\beta$ be the $x$-intercept of the tangent line to ellipse $C_{2}$ at point Q. How many values of $n$ satisfy $6 \leq \alpha - \beta \leq 15$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

If the asymptotes of the hyperbola $y ^ { 2 } - \frac { x ^ { 2 } } { m ^ { 2 } } = 1 ( m > 0 )$ are tangent to the circle $x ^ { 2 } + y ^ { 2 } - 4 y + 3 = 0$, then $m =$ $\_\_\_\_$

11. Let $O$ be the origin. Point $A ( 1,1 )$ lies on the parabola $C : x ^ { 2 } = 2 p y$ ( $p > 0$ ). A line through point $B ( 0 , - 1 )$ intersects $C$ at points $P$ and $Q$. Then
A. The directrix of $C$ is $y = - 1$
B. Line $A B$ is tangent to $C$
C. $| O P | \cdot | O Q | > | O A | ^ { 2 }$
D. $| B P | \cdot | B Q | > | B A | ^ { 2 }$

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 $y = x + c _ { 1 } , y = x + c _ { 2 }$ 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) $1 / 4$.

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 $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3}x + 2y = 4$, then which of the following options is the only CORRECT combination?
[A] (IV) (iii) (S)
[B] (IV) (iv) (S)
[C] (II) (iii) (R)
[D] (II) (iv) (R)

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
(A) $1 < e < \sqrt{2}$
(B) $\sqrt{2} < e < 2$
(C) $\Delta = a^{4}$
(D) $\Delta = b^{4}$

Let $P$ be a point on the parabola $y ^ { 2 } = 4 a x$, where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $( a , m )$ is
(A) $( 2,3 )$
(B) $( 1,3 )$
(C) $( 2,4 )$
(D) $( 3,4 )$

Consider the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $S ( p , q )$ be a point in the first quadrant such that $\frac { p ^ { 2 } } { 9 } + \frac { q ^ { 2 } } { 4 } > 1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle O R T$ is $\frac { 3 } { 2 }$, then which of the following options is correct?
(A) $q = 2 , p = 3 \sqrt { 3 }$
(B) $q = 2 , p = 4 \sqrt { 3 }$
(C) $q = 1 , p = 5 \sqrt { 3 }$
(D) $q = 1 , p = 6 \sqrt { 3 }$

Statement 1: $y = m x - \frac { 1 } { m }$ is always a tangent to the parabola, $y ^ { 2 } = - 4 x$ for all non-zero values of $m$. Statement 2: Every tangent to the parabola, $y ^ { 2 } = - 4 x$ will meet its axis at a point whose abscissa is nonnegative.
(1) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

The equation of the normal to the parabola, $x ^ { 2 } = 8 y$ at $x = 4$ is
(1) $x + 2 y = 0$
(2) $x + y = 2$
(3) $x - 2 y = 0$
(4) $x + y = 6$

The point of intersection of the normals to the parabola $y ^ { 2 } = 4 x$ at the ends of its latus rectum is :
(1) $( 0,2 )$
(2) $( 3,0 )$
(3) $( 0,3 )$
(4) $( 2,0 )$

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$, meets the $x$-axis and $y$-axis at $A$ and $B$, respectively. Then $OA ^ { 2 } - OB ^ { 2 }$, where $O$ is the origin, equals
(1) $- \frac { 20 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) 4
(4) $- \frac { 4 } { 3 }$

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is:
(1) $4x - 2y = 1$
(2) $4x + 2y = 7$
(3) $x + 2y = 4$
(4) $2y - x = 2$

The eccentricity of an ellipse whose centre is at the origin is $\frac { 1 } { 2 }$. If one of its directrices is $x = - 4$, then the equation of the normal to it at $\left( 1 , \frac { 3 } { 2 } \right)$ is:
(1) $4 x - 2 y = 1$
(2) $4 x + 2 y = 7$
(3) $x + 2 y = 4$
(4) $2 y - x = 2$

A hyperbola passes through the point $P ( \sqrt { 2 } , \sqrt { 3 } )$ and has foci at $( \pm 2 , 0 )$. Then the tangent to this hyperbola at $P$ also passes through the point:
(1) $( 3 \sqrt { 2 } , 2 \sqrt { 3 } )$
(2) $( 2 \sqrt { 2 } , 3 \sqrt { 3 } )$
(3) $( \sqrt { 3 } , \sqrt { 2 } )$
(4) $( - \sqrt { 2 } , - \sqrt { 3 } )$

The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is:
(1) $2y - x = 2$
(2) $4x - 2y = 1$
(3) $4x + 2y = 7$
(4) $x + 2y = 4$

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point
(1) $(3\sqrt{2}, 2\sqrt{3})$
(2) $(2\sqrt{2}, 3\sqrt{3})$
(3) $(\sqrt{3}, \sqrt{2})$
(4) $(-\sqrt{2}, -\sqrt{3})$