The locus of a point which divides the join of $A(-1, 1)$ and a variable point $B$ on the circle ${x^2} + {y^2} = 4$ in the ratio $3 : 2$ is:

$A$ rod $PQ$ of length $2a$ moves with its ends on the coordinate axes. Find the locus of the circumcenter of $\Delta OPQ$.

If the sum of the distances from a variable point $P$ to the given points $A(1,0)$ and $B(0,1)$ is $2$,then the locus of $P$ is

$A$ cow is tied to a corner (vertex) of a regular hexagonal fenced area of side $a \ m$ by a rope of length $\frac{5a}{2} \ m$ in a grass field. (The cow cannot graze inside the fenced area). What is the maximum possible area of the grass field to which the cow has access to graze?

$A$ stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$ and a point $P$ moving in the plane such that $PA = nPB$ $(n \neq 0)$. If $|n| \neq 1$,then the locus of point $P$ is....