The value of $a$,such that the power of the point $(1, 6)$ with respect to the circle $x^2 + y^2 + 4x - 6y - a = 0$ is $-16$,is

The equation of a circle touching the curve $|x - 1| + |y - 4| = 6$ is

The equation of the circle having the lines $x^2 + 2xy + 3x + 6y = 0$ as its normals and having a size just sufficient to contain the circle $x(x - 4) + y(y - 3) = 0$ is

The line $4x - 3y + 2 = 0$ intersects the circle $x^2 + y^2 - 2x + 6y + c = 0$ at two points $A$ and $B$,and the length of the chord $AB = 8$. If $(1, k)$ is a point on the given circle and $k > 0$,then $k =$

Circles $C_1$ and $C_2$,of radii $r$ and $R$ respectively,touch each other as shown in the figure. The line $l$,which is parallel to the line joining the centres of $C_1$ and $C_2$,is tangent to $C_1$ at $P$ and intersects $C_2$ at $A$ and $B$. If $R^2=2r^2$,then $\angle AOB$ equals

The straight line $2x - 3y = 1$ divides the circular region $x^2 + y^2 \leq 6$ into two parts. If $S = \left\{ \left(2, \frac{3}{4}\right), \left(\frac{5}{2}, \frac{3}{4}\right), \left(\frac{1}{4}, -\frac{1}{4}\right), \left(\frac{1}{8}, \frac{1}{4}\right) \right\}$,then the number of point$(s)$ in $S$ lying inside the smaller part is