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:

$A$ 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

The centroid of a variable triangle $ABC$ is at a distance of $5$ units from the origin. If $A = (2, 3)$ and $B = (3, 2)$,then the locus of $C$ is

$A$ point $P(x, y)$ is such that the sum of the squares of its distances from $(a, 0)$ and $(-a, 0)$ is $2b^2$. The equation representing the locus of $P$ 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 $ABC$?

$A$ circle cuts a chord of length $4a$ on the $x$-axis and passes through a point on the $y$-axis,distant $2b$ from the origin. Then the locus of the center of this circle is