The equation of the director circle of the circle ${x^2} + {y^2} = {a^2}$ is

The locus of the centroid of the triangle with vertices at $(a \cos \theta, a \sin \theta)$,$(b \sin \theta, -b \cos \theta)$ and $(1, 0)$ is (where $\theta$ is a parameter).

The number of pairs $(a, b)$ of positive real numbers satisfying $a^4+b^4 < 1$ and $a^2+b^2 > 1$ is

The locus of the midpoint of a chord of the circle $x^{2} + y^{2} = 4$ which subtends an angle of $45^{\circ}$ at the major arc of the circle is:

$A$ point moves such that its distance from the point $(-1, 0)$ is always three times its distance from the point $(0, 2)$. The locus of the point is

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: