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.

Let $C_1$ be the circle in the third quadrant of radius $3,$ that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches $C_1$ externally at the point $(\alpha, \beta)$. If $(\beta-\alpha)^2=\frac{m}{n}$,where $\operatorname{gcd}(m, n)=1$,then $m + n$ is equal to :

If two vertices of an equilateral triangle are $(-1, 0)$ and $(1, 0)$,then its circumcircle is:

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$,and given that $a - 2\sqrt{2}b = 3$,then $(a^{2} + b^{2} + ab)$ is equal to ..... .

The area of the circle having its centre at $(3,4)$ and touching the line $5x+12y-11=0$ is

What are the coordinates of points $E$ and $F$?