Two tangents to the circle $x^2+y^2=4$ at the points $A$ and $B$ meet at the point $P(-4,0)$. Then the area of the quadrilateral $PAOB$,where $O$ is the origin,is:

The equation of the normal at the point $(4, -1)$ of the circle $x^2 + y^2 - 40x + 10y = 153$ is

$A$ circle $C_{1}$ passes through the origin $O$ and has a diameter of $4$ on the positive $x$-axis. The line $y = 2x$ intersects the circle $C_{1}$ at $O$ and $A$. Let $C_{2}$ be the circle with $OA$ as a diameter. If the tangent to $C_{2}$ at the point $A$ meets the $x$-axis at $P$ and the $y$-axis at $Q$,then the ratio $QA : AP$ is equal to:

The point of contact of the tangent to the circle $x^2 + y^2 = 5$ at the point $(1, -2)$ which also touches the circle $x^2 + y^2 - 8x + 6y + 20 = 0$ is:

The line $3x - 2y = k$ meets the circle ${x^2} + {y^2} = 4{r^2}$ at only one point,if ${k^2} =$

The angle between the tangents drawn from a point $(4,3)$ to the circle $x^2+y^2-2x-4y=0$ is