If the point $(1, 4)$ lies inside the circle $x^2 + y^2 - 6x - 10y + p = 0$ and the circle does not touch or intersect the coordinate axes,then the set of all possible values of $p$ is the interval

If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.

The area of the triangle formed by joining the origin to the points of intersection of the line $x\sqrt{5} + 2y = 3\sqrt{5}$ and the circle $x^2 + y^2 = 10$ is

$A$ rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$,with two of its vertices on the line joining the centres of the circles. The area of the rhombus is:

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
$(A)$ $198$ $(B)$ $199$ $(C)$ $200$ $(D)$ $201$

The equation of a line inclined at an angle $\frac{\pi}{4}$ to the $X$-axis,such that the two circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 10x - 14y + 65 = 0$ intercept equal lengths on it,is