At which point on the line $x = 3$ are the tangents drawn to the circle $x^2 + y^2 = 8$ perpendicular to each other?

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2-4x+2y+3=0$ is $\frac{\pi}{2}$,then the locus of $P$ is

If the locus of the point,whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$,is $ax^2+by^2+cxy+dx+ey+170=0$,then the value of $a^2+2b+3c+4d+e$ is equal to:

The locus of the centre of the circle $\frac{1}{2} (x^2 + y^2) + x \cos \theta + y \sin \theta - 4 = 0$ is :-

The circle passing through the distinct points $(1, t)$,$(t, 1)$,and $(t, t)$ for all values of $t$ passes through the point:

If $P(x_1, y_1)$ is a point such that the lengths of the tangents from it to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ are in the ratio $2:3$,then the locus of $P$ is