The locus of a point which moves in such a way that its distance from $(0,0)$ is three times its distance from the $x$-axis is given by:

$A$ point $P$ moves such that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$. Then the locus of the point is

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B,$ then the locus of the foot of the perpendicular from $O$ on $AB$ is

If $P(x_1, y_1)$ is a point such that the lengths of the tangents from it to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ are in the ratio $2:3$,then the locus of $P$ is

If $P$ is a variable point such that the sum of the distances from $P$ to the points $A(2,2)$ and $B(2,-2)$ is $4$,then the locus of $P$ represents

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: