Two points and<img alt="\mathrm{ Q (-a,

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

Option: 1
$\mathrm{x^{2}+y^{2}-2xy\ cot \ 2x+a^{2}=0}$

Option: 2
$\mathrm{x^{2}+y^{2}+2xy\ cot \ \alpha -a^{2}=0}$

Option: 3
$\mathrm{x^{2}+y^{2}+2xy\ cot\ 2\alpha -a^{2}=0}$

Option: 4
None of these

Answers (1)

Let $\mathrm{R(h, k)}$ be the variable point (see figure). Then $\mathrm{\angle RPQ = \Theta }$ and $\mathrm{\angle RQP = \varphi , }$ so that $\mathrm{ \Theta -\varphi }$ $\mathrm{ = 2\alpha }$ . Let RM $\mathrm{ \perp PQ }$ , so that $\mathrm{ RM = k, }$

$\mathrm{ MP=a-h }$ and $\mathrm{ MQ = a + h. }$
Then $\mathrm{ tan \Theta = }$ $\mathrm{ \frac{RM}{MP}=\frac{k}{a-h} }$ and $\mathrm{tan\varphi =\frac{RM}{MQ}=\frac{k}{a+h}}$
$\mathrm{\therefore }$ from $\mathrm{2\alpha = \Theta - \alpha }$
$\mathrm{tan\ 2\alpha=tan \left ( \Theta -\varphi \right ) = }$ $\mathrm{\frac{tan \Theta -tan \phi }{1+tan\Theta \ tan\phi }=}$ $\mathrm{\frac{k\left ( a+h \right )-k\left ( a-h \right )}{a^{2}-h^{2}+k^{2}}}$