The lengths of tangents from a fixed point to three circles of a coaxial system are $t_1, t_2, t_3$. If $P, Q,$ and $R$ are the centers of these circles,then $QRt_1^2 + RPt_2^2 + PQt_3^2$ is equal to

Let $a=1+i$ and $z=x+iy$. If the curve $z\bar{z}+az+\bar{a}\bar{z}-4=0$ is cut by the straight line $(z+\bar{z})-i(z-\bar{z})+2=0$ at two points $A$ and $B$, then the equation of the circle passing through the origin, $A$ and $B$ is

If one of the two circles $x^2+y^2+\alpha_1(x-y)+c=0$ and $x^2+y^2+\alpha_2(x-y)+c=0$ lies within the other,then (where $\alpha_1, \alpha_2 \in R, \alpha_1 \neq \alpha_2$):

Number of circles intersecting $x^2+y^2=4$,$x^2+y^2-2x-3=0$ and $x^2+y^2-2y-3=0$ orthogonally is

$\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S_1: x^2+y^2-2x+6y=0$,$S_2: x^2+y^2+2gx-2y+6=0$,and $S_3: x^2+y^2-12x+2fy+3=0$. If $S_2$ and $S_3$ intersect orthogonally,then $(g, f) =$

If $C_1$ and $C_2$ are the centres of similitude with respect to the circles $x^2+y^2+6x+8y+24=0$ and $x^2+y^2-6x-8y+9=0$,then $C_1C_2=$