Two points $$P\left( {a,\,0} \right)$$   and $$Q\left( { - a,\,0} \right)$$   are given. $$R$$ is a variable point on one side of the line $$PQ$$  such that $$\angle RPQ - \angle RQP$$     is $$2\alpha .$$  Then, the locus of $$R$$ is :

Solution :
Let $$R\left( {h,\,k} \right)$$   be the variable point. Then,

$$\eqalign{ & \angle RPQ = \theta {\text{ and }}\angle RQP = \phi ,\,{\text{so that }}\theta - \phi = 2\alpha \cr & {\text{Let, }}RM \bot PQ,{\text{ so that}} \cr & RM = k,\,MP = a - h{\text{ and }}MQ = a + h \cr & {\text{Then, }}\tan \,\theta = \frac{{RM}}{{MP}} = \frac{k}{{a - h}}, \cr & \tan \,\phi = \frac{{RM}}{{MQ}} = \frac{k}{{a + h}} \cr & {\text{Therefore, from }}2\alpha = \theta - \phi ,\,{\text{we have}} \cr & \tan \,2\alpha = \tan \left( {\theta - \phi } \right) = \frac{{\tan \,\theta - \tan \,\phi }}{{1 + \tan \,\theta \,\tan \,\phi }} = \frac{{k\left( {a + h} \right) - k\left( {a - h} \right)}}{{{a^2} - {h^2} + {k^2}}} \cr & \Rightarrow {a^2} - {h^2} + {k^2} - 2hk\,\cot \,2\alpha = 0 \cr} $$
Therefore, the locus of $$R\left( {h,\,k} \right)$$   is $${x^2} - {y^2} + 2xy\,\cot \,2\alpha - {a^2} = 0$$
Hence, (A) is the correct answer.