$A$ light ray emerging from the point source placed at $P(1, 3)$ is reflected at a point $Q$ on the $x$-axis. If the reflected ray passes through the point $R(6, 7)$,then the abscissa of $Q$ is

$AB$ is a line segment moving between the axes such that '$A$' lies on $X$-axis and '$B$' lies on $Y$-axis. If $P$ is a point on $AB$ such that $PA=b$ and $PB=a$,then the equation of the locus of $P$ is

If a point $P$ on the line $3x + 5y = 15$ is equidistant from the coordinate axes,then $P$ lies

Let $P$ be a moving point such that the sum of its perpendicular distances from the lines $2x + y - 3 = 0$ and $x - 2y + 1 = 0$ is always $2 \, units$. Then,the area bounded by the locus of point $P$ is:

The locus of the mid-points of the points of intersection of $x \cos \theta + y \sin \theta = 1$ with the coordinate axes is

Let $A$ be the set of all points $(\alpha, \beta)$ such that the area of the triangle formed by the points $(5, 6), (3, 2),$ and $(\alpha, \beta)$ is $12 \text{ square units}.$ Then the least possible length of a line segment joining the origin to a point in $A$ is: