The Cartesian form of the polar equation $\theta = \tan^{-1} 2$ is

The equation of the locus of a point which is equidistant from the points $(2, 3)$ and $(4, 5)$ is

$A$ variable line passing through $(l, m)$ intersects the coordinate axes at the points $A$ and $B$. If the line drawn parallel to $Y$-axis through $A$ and parallel to $X$-axis through $B$ meet at $P$,then the locus of $P$ is

$A$ rod of length $2l$ slides with its ends on two perpendicular lines. The locus of its mid-point is

If the equation of a curve remains unchanged by replacing $x$ with $y$ and $y$ with $x$,then the curve is

$A$ line cuts the $x$-axis at $A(7, 0)$ and the $y$-axis at $B(0, -5)$. $A$ variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$-axis at $P(a, 0)$ and the $y$-axis at $Q(0, b)$. If $AQ$ and $BP$ intersect at $R(h, k)$,the locus of $R$ is