The centres of those circles which touch the circle $x^{2} + y^{2} - 8x - 8y - 4 = 0$ externally and also touch the $x$-axis,lie on:

The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are the tangents to the hyperbola $9x^2-16y^2=144$ is

$A$ point $P(x, y)$ is such that the sum of squares of its distances from the coordinate axes is equal to the square of its distance from the line $x-y=1$. Then the equation of the locus of $P$ is

If a circle $S$ passes through the origin and makes an intercept of length $4$ units on the line $x=2$,then the equation of the curve on which the centre of $S$ lies is

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

From a point $P$ on the circle $x^2+y^2=4$,two tangents are drawn to the circle $x^2+y^2-6x-6y+14=0$. If $A$ and $B$ are the points of contact of those lines,then the locus of the centre of the circle passing through the points $P, A$ and $B$ is