The lengths of the tangents from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-k=0$ are in the ratio $4:3$. Then the value of $k$ is:

The angle between the circles $x^2+y^2-2x-9=0$ and $x^2+y^2-4y-1=0$ at their point of intersection is

Let the line $x+y=1$ meet the circle $x^2+y^2=4$ at the points $A$ and $B$. If the line perpendicular to $AB$ and passing through the midpoint of the chord $AB$ intersects the circle at $C$ and $D$,then the area of the quadrilateral $ADBC$ is equal to

Let the tangents drawn from $P(-1, -1)$ to the circle $x^2 + y^2 - 2x - 4y - 4 = 0$ touch the circle at the points $A$ and $B$. Then the area of the triangle $PAB$ (in square units) is

Let $P(a_1, b_1)$ and $Q(a_2, b_2)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$,then $a_1^2 + a_2^2 + b_1^2 + b_2^2$ is equal to $...........$.

The minimum distance and maximum distance of the point $P(2,-7)$ from the circle $x^2+y^2-14x-10y-151=0$ are respectively . . . . . . units.