Question

Two points P and Q are given.R is a variable point on one side of the line PQ such that

$\angle RPQ$- $\angle RQP$ is a positive constant $2\alpha$. Find the locus of the point R.

Answer 1

The diagrammatic representation of the problem is shown below.

In $\Delta RMP$, $tan\theta =\frac{RM}{MP}=\frac{y_1}{a-x_1}$.

Also, in $\Delta RQM$, $tan\varphi =\frac{RM}{QM}=\frac{y_1}{a+x_1}$

Given $\angle RPQ-\angle RQP=2\alpha$

$\Rightarrow \theta -\alpha =2\alpha$

$\Rightarrow tan(\theta -\alpha )=tan\left(2\alpha \right)$

$\Rightarrow \frac{tan\theta -tan\alpha }{1+tan\theta tan\alpha }=tan\left(2\alpha \right)$

$\Rightarrow tan\left(2\alpha \right)=\frac{\frac{y_1}{a-x_1}-\frac{y_1}{a+x_1}}{1+\frac{y_1}{a-x_1}\times \frac{y_1}{a+x_1}}$

$\Rightarrow tan\left(2\alpha \right)=\frac{2x_1{y}_1}{a^2-x_1^2+y_1^2}$

$\Rightarrow a^2-x_1^2+y_1^2=2x_1{y}_{1}cot2\alpha$

$\Rightarrow x_1^2-y_1^2+2x_1{y}_{1}cot2\alpha =a^2$

Thus the locus of the point $R(x,y)$ is given by $x^2-y^2+2xycot2\alpha =a^2$