Let $x$ and $y$ be real numbers satisfying the equation $x^2 - 4x + y^2 + 3 = 0$. If the maximum and minimum values of $x^2 + y^2$ are $M$ and $m$ respectively,then the numerical value of $M - m$ is:

Consider a rigid square $ABCD$ as in the figure with $A$ and $B$ on the $X$ and $Y$-axes,respectively. When $A$ and $B$ slide along their respective axes,the locus of $C$ forms a part of

For any real number $t$,the point $\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)$ lies on a/an

$A$ 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?

The number of circles touching the line $y - x = 0$ and the $y$-axis is

If a circle $S$ passing through the point $(3,4)$ cuts the circle $x^2+y^2=36$ orthogonally,then the locus of the centre of $S$ is