The maximum possible value of $x^2+y^2-4x-6y$,where $x, y$ are real,subject to the condition $|x+y|+|x-y|=4$ is

$A$ chord $AB$ is drawn from the point $A(0,3)$ on the circle $x^{2}+4x+(y-3)^{2}=0$, and is extended to $M$ such that $AM=2AB$. The locus of $M$ is

$A$ point $P(x, y)$ divides the line segment joining the points $(5, 0)$ and $(10 \cos \theta, 10 \sin \theta)$ in the ratio $2 : 3$. Find the locus of point $P$ as $\theta$ varies.

The locus of the centre of a circle passing through $(a, b)$ and cutting orthogonally to the circle $x^2 + y^2 = p^2$ is

The locus of the midpoint of a chord of the circle $x^2 + y^2 = 4$ which subtends a right angle at the origin is

If the endpoints of the hypotenuse of a right-angled triangle are $(2, 0)$ and $(0, 2)$,find the locus of its third vertex.