The locus of the intersection point of $x \cos \alpha - y \sin \alpha = a$ and $x \sin \alpha - y \cos \alpha = b$ is

$A$ and $B$ are fixed points with coordinates $(0, a)$ and $(0, b)$ where $a > 0$ and $b > 0$. $P$ is a variable point $(x, 0)$ on the $x$-axis. If the angle $\angle APB$ is maximum, then:

The locus of a point,such that the difference of the squares of the lengths of the tangents drawn from it to two given circles is constant,is:

$A$ circle passes through the point $(3, 4)$ and cuts the circle $x^2 + y^2 = a^2$ orthogonally. The locus of its centre is a straight line. If the distance of this straight line from the origin is $25$,then $a^2$ is equal to:

The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin is:

$A$ 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