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

Let $AB$ be a chord of a circle and $C$ divides $AB$ internally in the ratio $3 : 1$. $A$ line through $C$ cuts the circle at $D$ and $E$ such that the minimum distances of $D$ and $E$ from line $AB$ are $3$ and $2$ respectively. If $r$ is the minimum length of $AB$ such that $\alpha$ is the angle between $AB$ and $DE$ for this $r$,then the value of '$r\alpha$' is

If the point $(2 \cos \theta, 2 \sin \theta)$ for $\theta \in (0, 2 \pi)$ lies in the region between the lines $x+y=2$ and $x-y=2$ containing the origin,then $\theta$ lies in

If $A(-1,3)$ and $B(5,3)$ are points on a circle $C$ and the chord $AB$ subtends an angle $\pi / 4$ at a point $P$ on $C$,then the equation of such a circle $C$ is

When do the two circles $x^2 + y^2 = ax$ and $x^2 + y^2 = c^2$ $(c > 0)$ touch each other?

$A$ line meets the circle $x^2+y^2-4x-4y-8=0$ at two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $PA=PB=2$,then the equation of the line $AB$ is: