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:

Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then,the number of such points $P$ in $S$ is

If the line $\frac{x}{a} + \frac{y}{b} = 1$ is a variable line such that $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$,then the locus of the foot of the perpendicular from the origin to the line is:

$A$ ray of light passing through the point $A(1, 2)$ is reflected at a point $B$ on the $x$-axis and then passes through $(5, 3)$. Then the equation of $AB$ is:

$A$ line passing through $P(4,2)$ cuts the coordinate axes at $A$ and $B$ respectively. If $O$ is the origin,then the locus of the centre of the circum-circle of $\triangle OAB$ is

The equation of the locus of the point of intersection of the straight lines $x \sin \theta + (1 - \cos \theta) y = a \sin \theta$ and $x \sin \theta - (1 + \cos \theta) y + a \sin \theta = 0$ is