The number of common tangents to two circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 8x + 12 = 0$ is

Three circles of radius $r$ touch each other externally. What is the radius of the circle that touches all three circles internally?

The angle of intersection of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the circle $x^2 + y^2 = ab$ is

For the four circles $M, N, O$ and $P$,the following four equations are given:
Circle $M: x^2 + y^2 = 1$
Circle $N: x^2 + y^2 - 2x = 0$
Circle $O: x^2 + y^2 - 2x - 2y + 1 = 0$
Circle $P: x^2 + y^2 - 2y = 0$
If the centre of circle $M$ is joined with the centre of circle $N$,the centre of circle $N$ is joined with the centre of circle $O$,the centre of circle $O$ is joined with the centre of circle $P$,and lastly,the centre of circle $P$ is joined with the centre of circle $M$,then these lines form the sides of a:

If $(m_i, \frac{1}{m_i}), i = 1, 2, 3, 4$ are concyclic points,then the value of $m_1 \cdot m_2 \cdot m_3 \cdot m_4$ is

If a point $P(\alpha, \beta)$ on the line $y=1$ is such that the two distinct chords drawn on $x^2+y^2-\alpha x-y=0$ from $P$ are bisected by the $x$-axis,then