The perimeter of the locus of the point $P$ which divides the line segment $QA$ internally in the ratio $1:2$,where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$ is

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2+4x-6y+9 \sin^2 \alpha+13 \cos^2 \alpha=0$ is $2 \alpha$,then the equation of the locus of $P$ is

The centroid of a variable triangle $ABC$ is at a distance of $5$ units from the origin. If $A = (2, 3)$ and $B = (3, 2)$,then the locus of $C$ is

If $P(x, y)$ is a point such that the ratio of the squares of the lengths of the tangents from $P$ to the circles $x^2 + y^2 + 2x - 4y - 20 = 0$ and $x^2 + y^2 - 4x + 2y - 44 = 0$ is $2 : 3$,then the locus of $P$ is a circle with centre

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

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at points $A$ and $B$. Then the minimum value of $OA+OB$,where $O$ is the origin,is equal to