The equation $x^2 + y^2 + 4x + 6y + 13 = 0$ represents

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

If the ratio of the distances of a variable point $P$ from the point $(1, 1)$ and the line $x-y+2=0$ is $1: \sqrt{2}$,then the equation of the locus of $P$ is

The locus of the center of a variable circle which always touches two given circles externally is

The equation $r \cos \theta = 2 a \sin^2 \theta$ represents the curve

$A$ point $P(x, y)$ is such that the sum of squares of its distances from the coordinate axes is equal to the square of its distance from the line $x-y=1$. Then the equation of the locus of $P$ is