The locus of the middle points of chords of the circle $x^2 + y^2 - 2x - 6y - 10 = 0$ which pass through the origin is:

If $P(x, y)$ is a point such that the ratio of the squares of the lengths of the tangents from $P$ to the circles $x^2 + y^2 + 2x - 4y - 20 = 0$ and $x^2 + y^2 - 4x + 2y - 44 = 0$ is $2 : 3$,then the locus of $P$ is a circle with centre

$A$ 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 equation of the locus of the centroid of the triangle whose vertices are $(a \cos k, a \sin k)$,$(b \sin k, -b \cos k)$ and $(1, 0)$,where $k$ is a parameter,is

Two straight rods of lengths $2a$ and $2b$ move along the coordinate axes in such a way that their extremities are always concyclic. Then the locus of the centres of such circles is

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 $ABC$?