If the sum of the distances of a point from two perpendicular lines in a plane is $1$,find its locus.

Let $A(2,1)$ be a point and the equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $c$ is the distance of the point $A$ from the origin such that $a=bc$,then the locus of $P$ is

If the point $(a, a^2)$ lies inside the angle formed by the lines $y = \frac{x}{2}$ $(x > 0)$ and $y = 3x$ $(x > 0)$,then $a$ belongs to:

The locus of the point $P(x, y)$ such that the area of the $\triangle PAB$ is $7$,where $A(4, 5)$ and $B(-2, 3)$ are given points,is

The set of all points that forms a triangle of area $15$ sq units with the points $(1, -2)$ and $(-5, 3)$ lies on

$A$ point $P$ moves on the line $2x - 3y + 4 = 0$. If $Q(1, 4)$ and $R(3, -2)$ are fixed points,then the locus of the centroid of $\Delta PQR$ is a line