Question

If $ax+by=1$ is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then $a^2-b^2$ equals to

• A. $\frac{1}{a^2{e}^2}$
• B. $a^2{e}^2$
• C. $b^2{e}^2$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{a^2{e}^2}$

Equation of tangent to the hyperbola is
$\frac{x}{a}\sec \theta -\frac{y}{b}\tan \theta =1$ .....(i)
Given tangent is,
$ax+by=1$ .....(ii)
Comparing Eqs. (i) and (ii), we have
$\sec \theta =a^2$ and $\tan \theta =-b^2$
$\Rightarrow a^4-b^4=1$
$\Rightarrow (a^2-b^2)(a^2+b^2)=1$
Also $a^2+b^2=a^2{e}^2$
$\Rightarrow a^2-b^2=\frac{1}{a^2{e}^2}$