$A$ point $P$ moves in such a way that the ratio of its distance from two coplanar points is always a fixed number $(\lambda \ne 1)$. Then its locus is

The angle between the pair of tangents 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 $P$ is...

If $A(2, 3)$ and $B(3, -2)$ are two fixed points and $P(x, y)$ is a variable point satisfying the condition $|PA - PB| = 2$,then the locus of $P$ is

The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin is:

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P$ moves in the plane such that $PA = nPB$ $(n \neq 0, n \neq 1)$. If $n > 1$,then which of the following is true regarding the positions of $A$ and $B$ with respect to the locus of $P$?

The locus of the point which is equidistant from the point $(1,1)$ and the line $x+y+1=0$ is