The point $P(10, 7)$ lies outside the circle $x^2 + y^2 - 4x - 2y - 20 = 0$. The greatest distance of $P$ from the circle is

$A$ triangle $PQR$ is inscribed in the circle $x^2 + y^2 = 25$. If the coordinates of $Q$ and $R$ are $(3, 4)$ and $(-4, 3)$ respectively,then $\angle QPR = \dots$

The line $l : x + y = 4$ intersects the circle $x^2 + y^2 - 2x - 2y = 2$ at points $A$ and $B$. If $C$ is the center of the circle, then the area of $\triangle ABC$ is...

When do the circles represented by the equations $x^{2} + y^{2} + c^{2} = 2ax$ and $x^{2} + y^{2} + c^{2} = 2by$ touch each other externally?

The square of the length of the tangent from the point $(3, -4)$ to the circle $x^2 + y^2 - 4x - 6y + 3 = 0$ is ....

Consider a semicircle of radius $1$ unit constructed on the diameter $AB$ and let $O$ be its centre. Let $C$ be a point on $AO$ such that $AC:CO = 2:1$. Draw $CD$ perpendicular to $AO$ with $D$ on the semicircle. Draw $OE$ perpendicular to $AD$ with $E$ on $AD$. Let $OE$ and $CD$ intersect at $H$. Then,$DH$ equals