The image of the point $(3, 4)$ with respect to the radical axis of the circles $x^2 + y^2 + 8x + 2y + 10 = 0$ and $x^2 + y^2 + 7x + 3y + 10 = 0$ is

The area of the circle passing through the points $(5, 2), (5, -2),$ and $(1, 2)$ is (in $\pi$)

$A$ rectangle is inscribed in a circle with a diameter lying along the line $3y = x + 7$. If the two adjacent vertices of the rectangle are $(-8, 5)$ and $(6, 5)$,then the area of the rectangle (in $sq. units$) is

If $\theta$ is the angle between the tangents from $(-1, 0)$ to the circle $x^2+y^2-5x+4y-2=0$,then $\theta$ is equal to

The radius of a circle $C_1$ is thrice the radius of another circle $C_2$. The centres of $C_1$ and $C_2$ are $(1, 2)$ and $(3, -2)$ respectively. If they cut each other orthogonally,find the equation of the circle with radius $r$ and centre $(1, -2)$.

Find the equation of the director circle of the circle $x^2 + y^2 = 8$.