If the circles $x^2+y^2+2hx+2ky=0$ and $x^2+y^2+2h'x+2k'y=0$ touch each other,then $\frac{h'k}{hk'} = $

$A$ rectangle is inscribed in a circle with a diameter lying along the line $3y = x + 7$. If the two adjacent vertices of the rectangle are $(-8, 5)$ and $(6, 5)$,then the area of the rectangle (in $sq. units$) is

Let a circle $C_1$ be obtained by rolling the circle $x^2+y^2-4x-6y+11=0$ upwards $4$ units along the tangent $T$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively,and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium $AMNB$ is:

$S_1$ and $S_2$ are two concentric circles of radii $1$ and $2$ respectively. Two parallel tangents to $S_1$ cut off an arc from $S_2$. The length of the arc is

If the circle $(x - h)^2 + (y - k)^2 = r^2$ touches the curve $y = x^2 + 1$ at the point $(1, 2)$,then the possible locations of the points $(h, k)$ are given by

The number of common tangents to the circles $x^2 + y^2 - 2x - 1 = 0$ and $x^2 + y^2 - 2y - 7 = 0$ is: