$A$ line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point,that divides the line segment $AB$ in the ratio $2:3$,is a circle of radius

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) $TRUE$?

Find the locus of a point such that the sum of the squares of its distances from the axes is $4$.

The radius of the circle with minimum area that touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is:

For a real variable $a > 1$,consider the points $A_k = (k a, a^k)$,$k = 1, 2, \ldots, n$ in the Cartesian plane. If $\alpha$ and $\beta$ represent respectively the arithmetic mean of $x$-coordinates and the geometric mean of $y$-coordinates of $A_k$,then the locus of the point $P(\alpha, \beta)$ is

The locus of the points from which perpendicular tangents can be drawn to the circle $x^2 + y^2 = a^2$ is