The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 20 = 0$ is

$A$ circle $C$ of radius $1$ is inscribed in an equilateral triangle $PQR$. The points of contact of $C$ with the sides $PQ, QR, RP$ are $D, E, F$,respectively. The line $PQ$ is given by the equation $\sqrt{3}x + y - 6 = 0$ and the point $D$ is $\left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right)$. Further,it is given that the origin and the centre of $C$ are on the same side of the line $PQ$.
$1.$ The equation of circle $C$ is
$(A) (x - 2\sqrt{3})^2 + (y - 1)^2 = 1$
$(B) (x - 2\sqrt{3})^2 + (y + \frac{1}{2})^2 = 1$
$(C) (x - \sqrt{3})^2 + (y + 1)^2 = 1$
$(D) (x - \sqrt{3})^2 + (y - 1)^2 = 1$
$2.$ Points $E$ and $F$ are given by
$(A) \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right), (\sqrt{3}, 0)$
$(B) \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), (\sqrt{3}, 0)$
$(C) \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right), \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$(D) \left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right), \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$3.$ Equation of the sides $QR, RP$ are
$(A) y = \frac{2}{\sqrt{3}}x + 1, y = -\frac{2}{\sqrt{3}}x - 1$
$(B) y = \frac{1}{\sqrt{3}}x, y = 0$
$(C) y = \frac{\sqrt{3}}{2}x + 1, y = -\frac{\sqrt{3}}{2}x - 1$
$(D) y = \sqrt{3}x, y = 0$
Give the answer for questions $1, 2$ and $3$.

Statement $1$: The only circle having radius $\sqrt{10}$ and a diameter along the line $2x + y = 5$ is $x^2 + y^2 - 6x + 2y = 0$.
Statement $2$: $2x + y = 5$ is a normal to the circle $x^2 + y^2 - 6x + 2y = 0$.

The length of the diameter of a circle which touches the $x$-axis at the point $(1, 0)$ and passes through the point $(2, -3)$ is:

The figure shows $\Delta ABC$ with $AB = 3, AC = 4$ and $BC = 5$. Three circles $S_1, S_2$ and $S_3$ have their centers at $A, B$ and $C$ respectively and they touch each other externally. The sum of the areas of the three circles 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 $(2)$: