$A$ variable straight-line $L$ with negative slope passes through the point $(4,9)$ and cuts the positive coordinate axes in $A$ and $B$. If $O$ is the origin, then the minimum value of $OA+OB$ 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

Let $BC$ be a fixed line segment in the plane. The locus of a point $A$ such that the $\triangle ABC$ is isosceles,is (with finitely many possible exceptional points)

If $A=(5,3)$,$B=(3,-2)$ and a point $P$ is such that the area of the triangle $PAB$ is $9$,then the locus of $P$ represents

If a variable line drawn through the point of intersection of straight lines $\frac{x}{\alpha} + \frac{y}{\beta} = 1$ and $\frac{x}{\beta} + \frac{y}{\alpha} = 1$ meets the coordinate axes in $A$ and $B$,then the locus of the midpoint of $AB$ is

$A$ point moves in the $xy$-plane such that the sum of its distances from two mutually perpendicular lines is always equal to $5$ units. The area (in sq units) enclosed by the locus of the point is