Let $A$ and $B$ be variable points on $x$-axis and $y$-axis respectively such that the line segment $AB$ is in the first quadrant and of a fixed length $2d$. Let $C$ be the mid-point of $AB$ and $P$ be a point such that
(a) $P$ and the origin are on the opposite sides of $AB$ and,
(b)$ PC$ is a line segment of length $d$ which is perpendicular to $AB$. Find the locus of $P$.
If we take points $A$ and $B$ as $(2d\cos\theta,0)$ and $(0,2d\sin\theta)$ respectively, then we get $C$ as $(d\cos\theta, d\sin\theta)$. Parametric eqn of $PC$ will be $$ \frac{x-d\cos \theta}{\sin\theta} = \frac{y-d\sin\theta}{\cos\theta}=d$$
Thus locus of $P$ is $x=y$ where $x,y \in [d,\sqrt2d]$
I wish to ensure the solution is correct and I didn't overlook anything.