The locus of the centre of circles passing through $(a, b)$ and cutting the circle $x^2+y^2-2x+4y-4=0$ orthogonally is

Find the locus of the midpoint of the chord of the circle $x^2 + y^2 = 16$ which is a tangent to the hyperbola $9x^2 - 16y^2 = 144$.

$A$ point $P(x, y)$ moves such that the sum of the squares of its distances from the points $(1, 2)$ and $(-2, 1)$ is $14$. Let $f(x, y) = 0$ be the locus of $P$,which intersects the $x$-axis at points $A, B$ and the $y$-axis at points $C, D$. Then the area of the quadrilateral $ACBD$ is equal to:

$A$ 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

If the distance of a variable point $P(x, y)$ from a point $A(2, -2)$ is twice the distance of $P$ from the $Y$-axis,then the equation of the locus of $P$ is:

$A$ variable circle passes through a fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter passing through $A$ is: