Question

Find the locus of the centre of circles which touch a given line at a given point.

Answer 1

Let $APB$ be the given line, and let a circle with centre $O$ touch $APB$ at $P$.

Then, $\angle OPB={90}^o$

Let there be another circle with centre $O^{\prime }$ which touches the line $APB$ at $P$.

Then, $\angle O^{\prime }PB={90}^o$

This is possible only when $O$ and $O^{\prime }$ lie on the same line $O^{\prime }OP$.

Hence the required locus is a line perpendicular to the given line at the point of contact.