The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are tangents to the hyperbola $9x^2-16y^2=144$ is

Three distinct points $A, B$,and $C$ are given in a two-dimensional coordinate plane such that for each point,the ratio of its distance from $(1, 0)$ to its distance from $(-1, 0)$ is equal to $\frac{1}{3}$. What is the circumcenter of triangle $ABC$?

If $A(-a, 0)$ and $B(a, 0)$ are two fixed points,then the locus of the point $P(x, y)$ on which the line segment $AB$ subtends a right angle is:

Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0)$. If the locus of the point $P$ is a circle of radius $r$,then $4r^{2}$ is equal to ...... .

The sum of the squares of the distances of a moving point from two fixed points $A(a, 0)$ and $B(-a, 0)$ is equal to a constant $2c^2$. Then,the equation of its locus is:

$A$ circle touches the $x$-axis and cuts off a chord of length $2l$ from the $y$-axis. The locus of the centre of the circle is