$A$ circle touches the $x$-axis and also touches the circle with centre at $(0, 3)$ and radius $2$. The locus of the centre of the circle is

If a circle passes through the point $(a, b)$ and cuts the circle $x^2 + y^2 = K^2$ orthogonally,then the equation of the locus of its centre is:

If $A(1, 1)$,$B(-1, 1)$,and $C(-1, -1)$ are three points and a point $P(x, y)$ moves such that $PA^2 = PB^2 + PC^2$,then the equation of the locus of $P$ is:

$A$ 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

Let $ABCD$ and $AEFG$ be squares of side $4$ and $2$ units,respectively. The point $E$ is on the line segment $AB$ and the point $F$ is on the diagonal $AC$. Then the radius $r$ of the circle passing through the point $F$ and touching the line segments $BC$ and $CD$ satisfies:

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: