The equation of radical axis of the circles ${x^2} + {y^2} + x – y + 2 = 0$ and $3{x^2} + 3{y^2} – 4x – 12 = 0,$ is

If the circles ${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} – 3ax + dy – 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5x + by – a = 0$ passes through $P$ and $Q$ for

Consider the circles ${x^2} + {(y – 1)^2} = $ $9,{(x – 1)^2} + {y^2} = 25$. They are such that

The equation of the image of the circle ${x^2} + {y^2} + 16x – 24y + 183 = 0$ by the line mirror $4x + 7y + 13 = 0$ is

If the variable line $3 x+4 y=\alpha$ lies between the two circles $(x-1)^{2}+(y-1)^{2}=1$ and $(x-9)^{2}+(y-1)^{2}=4$ without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is …. .