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
$y=mx\pm \sqrt{a^2{m}^2-b^2}$
$\Rightarrow {\left(k^2-mh\right)}^2=\left(a^2{m}^2-b^2\right)$
$\Rightarrow \left(h^2-a^2\right)m^2-2khm+k^2+b^2=0$
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 $x^2-y^2=a^2+b^2$