If the line $y = mx + 1$ meets the circle $x^2 + y^2 + 3x = 0$ in two points equidistant from and on opposite sides of the $x$-axis,then

$A$ line $y=mx+1$ intersects the circle $(x-3)^2+(y+2)^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$,then which one of the following options is correct?

$A$ line drawn through the point $A(5,7)$ cuts the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then,$AP \cdot AQ=$

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 length of the tangent drawn from the point $(-2, 3)$ to the circle $x^2 + y^2 + 8x - 6y + k = 0$ is $4$ units,then $k$ is equal to

The curve $xy = c, (c > 0)$,and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contact is