16. The focal chord to $y ^ { 2 } = 16 x$ is tangent to $( x - 6 ) ^ { 2 } + y ^ { 2 } = 2$, then the possible values of the slope of this chord, are :
(a) $\{ - 1,1 \}$
(b) $\{ - 2,2 \}$
(c) $\{ - 2,1 / 2 \}$
(d) $\{ 2,1 / 2 \}$

11. Find the equation of the common tangent in 1st quadrant to the circle $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipse $x ^ { 2 } / 25 + y ^ { 2 } / 4 = 1$. Also find the length of the intercept of the tangent between the coordinate axes.

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
A) $2 x - \sqrt { 5 } y - 20 = 0$
B) $2 x - \sqrt { 5 } y + 4 = 0$
C) $3 x - 4 y + 8 = 0$
D) $4 x - 3 y + 4 = 0$

If the normals of the parabola $y ^ { 2 } = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$, then the value of $r ^ { 2 }$ is

Suppose that the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ are $\left( f _ { 1 } , 0 \right)$ and $\left( f _ { 2 } , 0 \right)$ where $f _ { 1 } > 0$ and $f _ { 2 } < 0$. Let $P _ { 1 }$ and $P _ { 2 }$ be two parabolas with a common vertex at ( 0,0 ) and with foci at ( $f _ { 1 } , 0$ ) and ( $2 f _ { 2 } , 0$ ), respectively. Let $T _ { 1 }$ be a tangent to $P _ { 1 }$ which passes through ( $2 f _ { 2 } , 0$ ) and $T _ { 2 }$ be a tangent to $P _ { 2 }$ which passes through $\left( f _ { 1 } , 0 \right)$. If $m _ { 1 }$ is the slope of $T _ { 1 }$ and $m _ { 2 }$ is the slope of $T _ { 2 }$, then the value of $\left( \frac { 1 } { m _ { 1 } ^ { 2 } } + m _ { 2 } ^ { 2 } \right)$ is

Let $E _ { 1 }$ and $E _ { 2 }$ be two ellipses whose centers are at the origin. The major axes of $E _ { 1 }$ and $E _ { 2 }$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 2$. The straight line $x + y = 3$ touches the curves $S , E _ { 1 }$ and $E _ { 2 }$ at $P , Q$ and $R$, respectively. Suppose that $P Q = P R = \frac { 2 \sqrt { 2 } } { 3 }$. If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of $E _ { 1 }$ and $E _ { 2 }$, respectively, then the correct expression(s) is(are)
(A) $e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } = \frac { 43 } { 40 }$
(B) $\quad e _ { 1 } e _ { 2 } = \frac { \sqrt { 7 } } { 2 \sqrt { 10 } }$
(C) $\left| e _ { 1 } ^ { 2 } - e _ { 2 } ^ { 2 } \right| = \frac { 5 } { 8 }$
(D) $e _ { 1 } e _ { 2 } = \frac { \sqrt { 3 } } { 4 }$

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) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
[A] (I) (i) (P)
[B] (I) (ii) (Q)
[C] (II) (ii) (Q)
[D] (III) (i) (P)

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) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?
[A] (I) (ii) (Q)
[B] (II) (iv) (R)
[C] (III) (i) (P)
[D] (III) (ii) (Q)

Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ and the parabola $y ^ { 2 } = 4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O ( 0,0 )$ and whose semi-major axis is $O Q$. If the length of the minor axis of this ellipse is $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE?
(A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1
(B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$
(C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$
(D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is
$$\frac { 1 } { 16 } ( \pi - 2 )$$

Let the point $B$ be the reflection of the point $A ( 2,3 )$ with respect to the line $8 x - 6 y - 23 = 0$. Let $\Gamma _ { A }$ and $\Gamma _ { B }$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma _ { A }$ and $\Gamma _ { B }$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $A C$ is $\_\_\_\_$

Let $E$ denote the parabola $y ^ { 2 } = 8 x$. Let $P = ( - 2,4 )$, and let $Q$ and $Q ^ { \prime }$ be two distinct points on $E$ such that the lines $P Q$ and $P Q ^ { \prime }$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE ?
(A) The triangle $P F Q$ is a right-angled triangle
(B) The triangle $Q P Q ^ { \prime }$ is a right-angled triangle
(C) The distance between $P$ and $F$ is $5 \sqrt { 2 }$
(D) $F$ lies on the line joining $Q$ and $Q ^ { \prime }$

Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$

Let $T _ { 1 }$ and $T _ { 2 }$ be two distinct common tangents to the ellipse $E : \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 3 } = 1$ and the parabola $P : y ^ { 2 } = 12 x$. Suppose that the tangent $T _ { 1 }$ touches $P$ and $E$ at the points $A _ { 1 }$ and $A _ { 2 }$, respectively and the tangent $T _ { 2 }$ touches $P$ and $E$ at the points $A _ { 4 }$ and $A _ { 3 }$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 35 square units
(B) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 36 square units
(C) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 3,0 )$
(D) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 6,0 )$

Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is

Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$

Let $S$ denote the locus of the point of intersection of the pair of lines
$$\begin{gathered} 4 x - 3 y = 12 \alpha \\ 4 \alpha x + 3 \alpha y = 12 \end{gathered}$$
where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $( p , 0 )$ and $( 0 , q ) , q > 0$, and parallel to the line $4 x - \frac { 3 } { \sqrt { 2 } } y = 0$.
Then the value of $p q$ is

(A)

$- 6 \sqrt { 2 }$

(B)

$- 3 \sqrt { 2 }$

(C)

$- 9 \sqrt { 2 }$

(D)

$- 12 \sqrt { 2 }$

The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(1) $( - 1,1 )$
(2) $( 0,2 )$
(3) $( 2,4 )$
(4) $( - 2,0 )$

The number of common tangents of the circles given by $x ^ { 2 } + y ^ { 2 } - 8 x - 2 y + 1 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y = 0$ is
(1) one
(2) four
(3) two
(4) three

Given: A circle, $2x^2 + 2y^2 = 5$ and a parabola, $y^2 = 4\sqrt{5}x$. Statement-I: An equation of a common tangent to these curves is $y = x + \sqrt{5}$. Statement-II: If the line, $y = mx + \frac{\sqrt{5}}{m}$ $(m \neq 0)$ is their common tangent, then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

The number of common tangents to the circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 + 6x + 18y + 26 = 0$, is:
(1) 1
(2) 2
(3) 3
(4) 4

The number of common tangents to the circles $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0$, is
(1) 4
(2) 1
(3) 2
(4) 3

Equation of the tangent to the circle, at the point $( 1 , - 1 )$, whose center, is the point of intersection of the straight lines $x - y = 1$ and $2 x + y = 3$ is:
(1) $x + 4 y + 3 = 0$
(2) $3 x - y - 4 = 0$
(3) $x - 3 y - 4 = 0$
(4) $4 x + y - 3 = 0$

If the common tangents to the parabola, $x ^ { 2 } = 4 y$ and the circle, $x ^ { 2 } + y ^ { 2 } = 4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is:
(1) $2 ( 3 + 2 \sqrt { 2 } )$
(2) $3 + 2 \sqrt { 2 }$
(3) $\sqrt { 2 } + 1$
(4) $2 ( \sqrt { 2 } + 1 )$

If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is:
(1) 95
(2) 195
(3) 185
(4) 85

Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $x + 2 y + 3 = 0$
(4) $4 ( x + y ) + 3 = 0$