$A$ chord $AB$ is drawn from the point $A(0,3)$ on the circle $x^2 + 4x + (y - 3)^2 = 0$. If the point $M$ lies on the chord such that $AM = 2AB$,then the locus of point $M$ is:

If a circle passes through the point $(1, 2)$ and intersects the circle $x^2 + y^2 = 4$ orthogonally,then the equation of the locus of its center is:

The locus of the centres of all circles which touch the line $x=2a$ and cut the circle $x^2+y^2=a^2$ orthogonally is:

What does the equation $\sqrt{(x - 2)^2 + (y + 1)^2} + \sqrt{(x + 2)^2 + (y + 4)^2} = 5$ represent?

Let $P(\alpha, \beta)$ be a variable point which moves in the $x-y$ plane such that $\frac{PA}{PB} = 2$,where $A(1, 0)$ and $B(0, -1)$. If $M$ and $m$ denote respectively the maximum and minimum value of $\alpha + \beta$,then the value of $[\frac{M}{m}]$ is- (where $[.]$ denotes the greatest integer function)

$A$ point $P(x, y)$ is such that the sum of the squares of its distances from $(a, 0)$ and $(-a, 0)$ is $2b^2$. The equation representing the locus of $P$ is