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 $a = \sqrt { \mathbf { Y } }$.

A tangent to the circle $x ^ { 2 } + y ^ { 2 } = 144$ passes through the point $( 20,0 )$ and crosses the positive $y$-axis.
What is the value of $y$ at the point where the tangent meets the $y$-axis?
A 12
B 15
C $\frac { 49 } { 3 }$
D 20
E $\frac { 64 } { 3 }$
F $\frac { 80 } { 3 }$

The circle $C_1$ has equation $(x+2)^2 + (y-1)^2 = 3$
The circle $C_2$ has equation $(x-4)^2 + (y-1)^2 = 3$
The straight line $l$ is a tangent to both $C_1$ and $C_2$ and has positive gradient.
The acute angle between $l$ and the $x$-axis is $\theta$
Find the value of $\tan\theta$
A $\frac{1}{2}$
B $2$
C $\frac{\sqrt{2}}{2}$
D $\sqrt{2}$
E $\frac{\sqrt{6}}{2}$
F $\frac{\sqrt{6}}{3}$
G $\frac{\sqrt{3}}{3}$
H $\sqrt{3}$

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) $\sqrt { 7 }$