If the point $(x, y)$ is equidistant from the points $(a + b, b - a)$ and $(a - b, a + b)$,then:

The point moves such that the area of the triangle formed by it with the points $(1, 5)$ and $(3, -7)$ is $21$ sq. units. The locus of the point is:

$A$ point $P$ on a line is at a distance of $4$ units from the origin $(0,0)$. If the line makes an angle of $60^{\circ}$ with the negative direction of the $X$-axis,then the coordinates of $P$ are

Let $A=(4,0)$ and $B=(0,12)$ be two points in the plane. The locus of a point $C(x, y)$ such that the area of $\triangle ABC$ is $18$ sq units is

Let $A=(1, 2)$,$B=(2, 1)$,and $C=(-1, -1)$ be three points. If $P(x, y)$ is a point such that the area of the quadrilateral $PABC$ is twice the area of the triangle $PAB$,then the equation of the locus of $P$ is:

$A$ variable line passes through the fixed point $(\alpha, \beta)$. The locus of the foot of the perpendicular from the origin on the line is