$A$ circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center is

Let $A = (0, 4)$ and $B = (2 \cos \theta, 2 \sin \theta)$,for some $0 < \theta < \frac{\pi}{2}$. Let $P$ divide the line segment $AB$ in the ratio $2:3$ internally. The locus of $P$ is

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

Three distinct points $A, B, C$ are given in a two-dimensional coordinate plane such that the ratio of the distance of each point from $(1, 0)$ to its distance from $(-1, 0)$ is equal to $\frac{1}{2}$. What is the circumcenter of triangle $ABC$?

The locus of the points on the curve ${y^2 = 4a(x + a \sin \frac{x}{a})}$ where the tangent is parallel to the $x$-axis represents . . . .

The locus of the centre of the circles which touch both the circles $x^{2}+y^{2}=a^{2}$ and $x^{2}+y^{2}=4ax$ externally is