Question

From a point $P,$ two tangents $PA$ and $PB$ are drawn to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.$ If these tangents cut the coordinate axes at $4$ concyclic points, then the locus of $P$ is

• A. $x^{2}-y^{2}=\left|a^{2} - b^{2}\right|$
• B. $x^{2}-y^{2}=a^{2}+b^{2}$
• C. $x^{2}+y^{2}=\left|a^{2} - b^{2}\right|$
• D. $x^{2}+y^{2}=a^{2}+b^{2}$

Answer 1

Answer: (B)

Let the equation of tangents are
$y=m_{1}x+C_{1},$ and
$y=m_{2}x+C_{2}$
which cuts the coordinate axes at $E,F,G,H$ as shown in the figure
Now, $OE\times OG=OF\times OH$
$\Rightarrow \left(\frac{- C_{1}}{m_{1}}\right)\times \left(\frac{+ C_{2}}{m_{2}}\right)=\left(C_{1}\right)\left(- C_{2}\right)\Rightarrow m_{1}m_{2}=1$
Let $P$ be $\left(h , k\right)$ and equation of the tangent through $P$ on the hyperbola is
$y=mx\pm\sqrt{a^{2} m^{2} - b^{2}}$
$\Rightarrow \left(k^{2} - m h\right)^{2}=\left(a^{2} m^{2} - b^{2}\right)$
$\Rightarrow \left(h^{2} - a^{2}\right)m^{2}-2khm+k^{2}+b^{2}=0$
whose roots are $m_{1}$ and $m_{2}\Rightarrow m_{1}m_{2}=\frac{k^{2} + b^{2} }{h^{2} - a^{2}}=1$
$\Rightarrow $ locus is $x^{2}-y^{2}=a^{2}+b^{2}$