If $A$ and $B$ are two fixed points in a plane and $P$ is another variable point such that $PA^2 + PB^2 = k$ (where $k$ is a constant),then the locus of the point $P$ is:

The locus of a point $P(x, y)$ satisfying the equation $\sqrt{(x-2)^2+y^2}+\sqrt{(x+2)^2+y^2}=4$ is:

The locus of the midpoints of the chords of the circle $C_1: (x-4)^2 + (y-5)^2 = 4$ which subtend an angle $\theta_i$ at the centre of the circle $C_1$,is a circle of radius $r_i$. If $\theta_1 = \frac{\pi}{3}$,$\theta_3 = \frac{2\pi}{3}$ and $r_1^2 = r_2^2 + r_3^2$,then $\theta_2$ is equal to

The equation of the locus of a point $(x, y)$ which is at a distance of $5$ units from a fixed point $(1, 4)$ and also at a distance of $5$ units from a fixed line $2x + 3y - 1 = 0$ is:

Let $C$ be a circle with radius $3$ and center at $(0, 0)$. The equation of the locus of the midpoint of the chord of the circle that subtends an angle of $\frac{2\pi}{3}$ at the center is:

Let $S = 0$ be the locus of the center of a variable circle which intersects the circle $x^2 + y^2 - 4x - 6y = 0$ orthogonally at the point $(4, 6)$. If $P$ is a variable point on $S = 0$,then the least value of $OP$ is (where $O$ is the origin).