The line $x+y=k$ meets the curve $x^2+y^2-2x-4y+2=0$ at two points $A$ and $B$. If $O$ is the origin and $\angle AOB=90^{\circ}$,then the value of $k$ $(k>1)$ is

The line through $P(a, 2)$,where $a \neq 0$,making an angle $45^{\circ}$ with the positive direction of the $X$-axis meets the curve $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A$ and $D$ and the coordinate axes at $B$ and $C$. If $PA, PB, PC$ and $PD$ are in a geometric progression,then $2a=$

If a circle passes through the points of intersection of the coordinate axes with the lines $\lambda x - y + 1 = 0$ and $x - 2y + 3 = 0$,then the value of $\lambda$ is

The set of all values of $a^2$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$ is equal to:

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

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to: