If $OA$ and $OB$ are the tangents to the circle $x^2 + y^2 - 6x - 8y + 21 = 0$ drawn from the origin $O$,then $AB =$

Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is:

If $OA$ and $OB$ are tangents from the origin to the circle $x^{2} + y^{2} - 6x - 8y + 21 = 0$,then $AB = \dots$

The point of intersection of the direct common tangents drawn to the circles $(x+11)^2+(y-2)^2=225$ and $(x-11)^2+(y+2)^2=25$ is

If the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ meets the circle at $A$ and $B$; and if $\angle AOB=90^{\circ}$,then $x_1^2+y_1^2=$

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius $2$. If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point $P$,then the square of the distance of $P$ from the centre of $C_1$ is: