The radius of the circle having its centre at $(0, 3)$ and passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

Let $a$ and $b$ be non-zero real numbers. Then,the equation $(a x^2+b y^2+c)(x^2-5 x y+6 y^2)=0$ represents

$A$ variable line $ax + by + c = 0$,where $a, b, c$ are in $A.P.$,is normal to a circle $(x - \alpha)^2 + (y - \beta)^2 = \gamma$,which is orthogonal to the circle $x^2 + y^2 - 4x - 4y - 1 = 0$. The value of $\alpha + \beta + \gamma$ is equal to

If a tangent is drawn from the focus of the parabola $y^2 = 16x$ to the circle $(x - 6)^2 + y^2 = 2$,find the slope of this tangent.

Let $M = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 \leq r^2\}$,where $r > 0$. Consider the geometric progression $a_n = \frac{1}{2^{n-1}}$,$n = 1, 2, 3, \ldots$. Let $S_0 = 0$ and,for $n \geq 1$,let $S_n$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$,let $C_n$ denote the circle with center $(S_{n-1}, 0)$ and radius $a_n$,and $D_n$ denote the circle with center $(S_{n-1}, S_{n-1})$ and radius $a_n$.
$(1)$ Consider $M$ with $r = \frac{1025}{513}$. Let $k$ be the number of all those circles $C_n$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
$(A)$ $k + 2l = 22$ $(B)$ $2k + l = 26$ $(C)$ $2k + 3l = 34$ $(D)$ $3k + 2l = 40$
$(2)$ Consider $M$ with $r = \frac{(2^{199}-1)\sqrt{2}}{2^{198}}$. The number of all those circles $D_n$ that are inside $M$ is
$(A)$ $198$ $(B)$ $199$ $(C)$ $200$ $(D)$ $201$

For the circle $C$ with the equation $x^2+y^2-16x-12y+64=0$,match the List-$I$ with the List-$II$ given below.

List-$I$	List-$II$
$(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$	$(A)$ $y = 0$
$(ii)$ The equation of the tangent at $(8, 0)$ to $C$	$(B)$ $y = 6$
$(iii)$ The equation of the normal at $(2, 6)$ to $C$	$(C)$ $x + y = 7$
$(iv)$ The equation of the diameter of $C$ through $(8, 12)$	$(D)$ $13x + 5y = 98$
	$(E)$ $x = 8$

The correct match is: