The equation of a common tangent to the parabolas $y = x ^ { 2 }$ and $y = -(x - 2) ^ { 2 }$ is
(1) $y = 4 x - 2$
(2) $y = 4 x - 1$
(3) $y = 4 x + 1$
(4) $y = 4 x + 2$

Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is
(1) $\frac { - 9 } { 4 }$
(2) $\frac { 9 } { 4 }$
(3) $\frac { - 5 } { 2 }$
(4) $- 3$

Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65

A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$

Two tangent lines $l _ { 1 }$ and $l _ { 2 }$ are drawn from the point $( 2,0 )$ to the parabola $2 y ^ { 2 } = - x$. If the lines $l _ { 1 }$ and $l _ { 2 }$ are also tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r$, then $17 r ^ { 2 }$ is equal to $\_\_\_\_$.

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ meet at the point $R ( \sqrt { 2 } , 2 \sqrt { 2 } - 2 )$. If $S$ is the focus of the ellipse on its negative major axis, then $S P ^ { 2 } + S Q ^ { 2 }$ is equal to $\_\_\_\_$.

The number of common tangents, to the circles $x ^ { 2 } + y ^ { 2 } - 18 x - 15 y + 131 = 0$ and $x ^ { 2 } + y ^ { 2 } - 6 x - 6 y - 7 = 0$, is
(1) 3
(2) 1
(3) 4
(4) 2

Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.

Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$

A circle with centre $( 2,3 )$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S ( \alpha , \beta )$, then $4 \alpha - 7 \beta$ is equal to $\_\_\_\_$

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^{2}$ is equal to
(1) 64
(2) 76
(3) 81
(4) 72

Let a common tangent to the curves $y^2 = 4x$ and $(x-4)^2 + y^2 = 16$ touch the curves at the points $P$ and $Q$. Then $PQ^2$ is equal to $\_\_\_\_$.

Equations of two diameters of a circle are $2 x - 3 y = 5$ and $3 x - 4 y = 7$. The line joining the points $\left( - \frac { 22 } { 7 } , - 4 \right)$ and $\left( - \frac { 1 } { 7 } , 3 \right)$ intersects the circle at only one point $P ( \alpha , \beta )$. Then $17 \beta - \alpha$ is equal to

Let $C$ be a circle with a radius of 4, centered at the point $( 5,0 )$ on the $x$-axis.
(1) If $\mathrm { P } ( p , q )$ is a point on circle $C$, then
$$p ^ { 2 } - \mathbf { PQ } p + q ^ { 2 } + \mathbf { R } = 0 .$$
Also, the equation of the tangent to circle $C$ at point $\mathrm { P } ( p , q )$ is
$$( p - \mathbf { S } ) x + q y = \mathbf { T } p - \mathbf { U } .$$
(2) Let us draw a line tangent to circle $C$ from point $\mathrm { A } ( 0 , a )$ on the $y$-axis, where $a \geqq 0$, and let $\mathrm { P } ( p , q )$ be the tangent point.
The length of the segment AP is minimized at $a = \mathbf { V }$, and the length in this case is $\mathbf { W }$.
Furthermore, the two tangents to circle $C$ from point A are orthogonal when the length of AP is $\mathbf { X }$. In this case, the value of $a$ is $a = \sqrt { \mathbf { Y } }$.

The method of drawing a tangent to a circle with center O from an external point P is given below.

• Line segment OP is drawn.
• The midpoint M of line segment OP is determined.
• A circle with center M and diameter [OP] is drawn.
• The intersection points of the circles with centers O and M are marked. Let one of these points be T.
• Ray [PT is tangent to the circle with center O at point T.

In this construction, if the radii of the circles with centers O and M are 4 cm and 3 cm respectively, what is the length $| PT |$ in cm?
A) $3 \sqrt { 3 }$
B) $2 \sqrt { 5 }$
C) $\sqrt { 7 }$