The locus of a point which moves so that its distance from the $x$-axis is double its distance from the $y$-axis is:

The locus of a point,such that the difference of the squares of the lengths of the tangents drawn from it to two given circles is constant,is:

Let $T'$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T'$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$,and also such that $S_1$ and $S_2$ touch each other at a point,say,$M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then,which of the following statement$(s)$ is (are) $TRUE$?

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x^2 + y^2 = 16$. If the centre of the locus of the point $P$,which divides the line segment $AB$ in the ratio $3:2$ is the point $C(\alpha, \beta)$,then the length of the line segment $AC$ is

The locus of the mid-points of the chords of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ which subtend $60^o$ at the centre 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