$A$ 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

Consider a pair of circles $(|x| - 1)^2 + y^2 = 1$. Ram is moving along the circle centered at $(1, 0)$ in the clockwise direction at a rate of $2 \ m/s$,and Shyam is moving along the circle centered at $(-1, 0)$ in the anticlockwise direction at a rate of $1 \ m/s$. If Ram and Shyam start their journey from the origin $(0, 0)$,then the rate of change of the distance between Ram and Shyam at the instant when Ram crosses the $x$-axis for the first time is:

$A$ line drawn through the point $P(4, 7)$ cuts the circle $x^2 + y^2 = 9$ at the points $A$ and $B$. Then $PA \cdot PB$ is equal to

If $P(2, 8)$ is an interior point of a circle $x^2 + y^2 - 2x + 4y - p = 0$ which neither touches nor intersects the axes,then the set for $p$ is

$A$ variable circle passes through the fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter through $A$ is

Let $A=\{(x, y) \in R \times R \mid 2 x^{2}+2 y^{2}-2 x-2 y=1\}$,$B=\{(x, y) \in R \times R \mid 4 x^{2}+4 y^{2}-16 y+7=0\}$ and $C=\{(x, y) \in R \times R \mid x^{2}+y^{2}-4 x-2 y+5 \leq r^{2}\}$. Then the minimum value of $|r|$ such that $A \cup B \subseteq C$ is equal to: