Question

Two different points $P$ and $Q$ are given on one side of line $PB$. Draw a circle passing through the points $P$ and $Q$ touching the line $AB$ in point $R$.

Answer 1

Let $QP$ be chosen along x-axis and its length $=2a$ and mid-point as origin so that $P$ is $(a,0)$ and $Q$ is $(-a,0)$. Let $R$ be $(x,y)$ and $\angle RPQ=\theta ;\angle RQP=\varphi$
$\therefore$ $\theta -\varphi =2\alpha$ given and RM be the perpendicular on x-axis
$\therefore \frac{\tan \theta -\tan \varphi }{1-\tan \theta \tan \varphi }=\tan 2\alpha ....\left(1\right)$
But $\tan \theta =\frac{RM}{MP}=\frac{y}{a-x}$
$\tan \varphi =\frac{RM}{MQ}=\frac{y}{a+x}$
Putting in (1) we get
$\frac{y/(a-x)-y/(a-x)}{1+\left[y/\right(a+x\left)\right].\left[y/\right(a-x\left)\right]}=\tan 2\alpha$
or $2xy\cot 2\alpha =a^2-x^2+y^2$
or $y^2-x^2-2xy\cot 2\alpha +a^2=0$ is the required locus