$A$ point moves in such a way that its distance from the origin is always $4$. Then the locus of the point is:

The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area 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

The angle between a pair of tangents drawn from a point $P$ to the circle ${x^2} + {y^2} + 4x - 6y + 9{\sin ^2}\alpha + 13{\cos ^2}\alpha = 0$ is $2\alpha$. The equation of the locus of the point $P$ is

The locus of the centre of the circles passing through the origin and cutting off a chord of length $2$ units on the line $x=1$ is

$A$ point $P(x, y)$ is such that its distances from $(-1, 0)$ and $(0, 2)$ are in the ratio $\sqrt{2} : 1$. Then the locus of $P$ is