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

The circumference of a circle passing through the point $(4,6)$ with two normals represented by $2x - 3y + 4 = 0$ and $x + y - 3 = 0$ is (in $\pi$)

Consider the region $R = \{( x , y ) \in \mathbb{R} \times \mathbb{R} : x \geq 0 \text{ and } y^2 \leq 4- x \}$. Let $F$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has the largest radius among the circles in $F$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^2=4- x$.
$(1)$ The radius of the circle $C$ is. . . . . .
$(2)$ The value of $\alpha$ is. . . . .
Given the answer for $(1)$ and $(2)$:

If the points $P(3, 1)$ and $Q(6, 5)$ form a triangle with a third point $R(x, y)$ such that the area of $\Delta PQR$ is $6$ square units and $\angle PRQ = \frac{\pi}{2}$,then the number of possible positions for point $R$ is:

The number of common tangents that can be drawn to the circles $x^2 + y^2 - 6x = 0$ and $x^2 + y^2 + 6x + 2y + 1 = 0$ is .....

The equation of the chord of the circle $x^2+y^2-4x=0$ whose midpoint is $(1,0)$ is