Radical axis of the circles $3{x^2} + 3{y^2} - 7x + 8y + 11 = 0$ and ${x^2} + {y^2} - 3x - 4y + 5 = 0$ is

Two circles whose radii are equal to $4$ and $8$ intersects at right angles. The length of  their common chord is:-

If $d$ is the distance between the centres of two circles, ${r_1},{r_2}$ are their radii and $d = {r_1} + {r_2}$, then

Consider a circle $C_1: x^2+y^2-4 x-2 y=\alpha-5$.Let its mirror image in the line $y=2 x+1$ be another circle $C _2: 5 x ^2+5 y ^2-10 fx -10 gy +36=0$.Let $r$ be the radius of $C _2$. Then $\alpha+ r$ is equal to $......$.

Let $C_1$ and $C_2$ be the centres of the circles $x^2 + y^2 -2x -2y -2 = 0$ and $x^2 + y^2 - 6x-6y + 14 = 0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $PC_1QC_2$ is ............. $\mathrm{sq. \, units}$

If $y = 2x$ is a chord of the circle ${x^2} + {y^2} - 10x = 0$, then the equation of the circle of which this chord is a diameter, is