Equation of Tangent in Point Form

Q.

Answer the following by appropriately matching the lists based on the information given in the paragraph.

Let the circles $C_1:x^2+y^2=9$and $C_2:{(x–3)}^2+{(y–4)}^2=16$, intersect at the points $X$ and $Y$.

Suppose that another circle $C_3:{(x–h)}^2+{(y–k)}^2=r^2$ satisfies the following conditions:

(i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$

(ii) $C_1$ and $C_2$ both lie inside $C_2$, and

(iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$

Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2=8\alpha y$.

There are some expressions given in the $List–I$ whose values are given in $List–II$ below:

$List-I$	$List-II$
$\left(I\right)2h+k$	$\left(P\right)6$
$\left(II\right)\frac{LengthofZW}{LengthofXY}$	$\left(Q\right)\sqrt{6}$
$\left(III\right)\frac{AreaoftriangleMZN}{AreaoftriangleZMW}$	$\left(R\right)\frac{5}{4}$
$\left(IV\right)\alpha$	$\left(S\right)\frac{21}{5}$
	$\left(T\right)2\sqrt{6}$
	$\left(U\right)\frac{10}{3}$

Which of the following is the only INCORRECT combination?

1. $\left(IV\right),\left(S\right)$

2. $\left(IV\right),\left(U\right)$

3. $\left(III\right),\left(R\right)$

4. $\left(I\right),\left(P\right)$

Q. The circle $C_1:x^2+y^2=8$ cuts orthogonally the circle $C_2$ whose centre lies on the line $x-y-4=0$ then, the circle $C_2$ passes through a fixed point, which lies on

1. $x-2y=0$
2. $x+y=0$
3. $x-2y=0$
4. $x+2y=0$

Q. If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

1. $x^2+y^2-2xy=0$
2. $x^2+y^2-2x^2{y}^2=0$
3. $x^2+y^2-4x^2{y}^2=0$
4. $x^2+y^2-16x^2{y}^2=0$

Q. If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

Q. If tangent to the circle $x^2+y^2=5$ at $(1,-2)$ also touches the circle $x^2+y^2-8x+6y+20=0$ at point $(h,k),$ then the value of $h+k$ is