Q. If the line $lx + my = 1$ is a normal to the hyperbola $\frac{x^{2}}{a^{2}}\ - \ \frac{y^{2}}{b^{2}} = 1$, then $\frac{a^{2}}{l^{2}} - \frac{b^{2}}{m^{2}}$ is equal to
A) $a^{2} - b^{2}$
B) $a^{2} + b^{2}$
C) $\left( a^{2} + b^{2} \right)^{2}$
D) $\left( a^{2} - b^{2} \right)^{2}$
Explanation
(c)
We know, if $lx + my + n = 0$ is normal to the hyperbola
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Then, $\frac{a^{2}}{l^{2}} - \frac{b^{2}}{m^{2}} = \frac{\left( a^{2} + b^{2} \right)^{2}\ }{n^{2}}\$
Put, $n = - 1$, therefore, $\frac{a^{2}}{l^{2}} - \frac{b^{2}}{m^{2}} = \left( a^{2} + b^{2} \right)^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.