If a chord is drawn from the origin to the circle $(x - 1)^2 + y^2 = 1$,then the equation of the locus of the midpoint of this chord 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 $P(\alpha, \beta)$ 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)

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

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2+4x-6y+9 \sin^2 \alpha+13 \cos^2 \alpha=0$ is $2 \alpha$,then the equation of the locus of $P$ is

At which point on the line $x = 3$ are the tangents drawn to the circle $x^2 + y^2 = 8$ perpendicular to each other?