Find the radius of a circle that touches the $y-$axis at point $P(0,2)$ and touches the circle $x^2 + y^2 = 16$ internally.

From a point $P$ on the line $4x - 3y = 6$,two tangents are drawn to the circle $x^2 + y^2 - 6x - 4y + 4 = 0$. If the angle between these tangents is $\tan^{-1}\left(\frac{24}{7}\right)$,then $P$ can be:

Let the circle with centre at origin pass through the vertices of an equilateral triangle $ABC$. If $A = (2, 4)$,then the length of the median through $A$ is

If the parametric values of two points $A$ and $B$ on the circle $x^2+y^2-6x+4y-12=0$ are $30^{\circ}$ and $90^{\circ}$ respectively,then the equation of chord $AB$ is

For what values of $m$ does the circle $x^2 + y^2 = 4x + 8y + 5$ intersect the line $3x - 4y = m$ at two distinct points?

The least and the greatest distances of the point $(10, 7)$ from the circle $x^{2} + y^{2} - 4x - 2y - 20 = 0$ are