Question

The equations to the common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

• A. $y=\pm x\pm \sqrt{b^2-a^2}$
• B. $y=\pm x\pm (a^2-b^2)$
• C. $y=\pm x\pm \sqrt{a^2-b^2}$
• D. $y=\pm x\pm \sqrt{a^2+b^2}$

Answer 1

Answer: (C)

$y=\pm x\pm \sqrt{a^2-b^2}$

Equation of tangents to two hyperbolas are
$y=mx\pm \sqrt{a^2{m}^2-b^2}$ ....(i)
$y=mx\pm \sqrt{-b^2{m}^2-a^2}$ ....(ii)
Solving (i) & (ii) we get m = $\pm$ 1
$\therefore$ equation of common tangent is
$y=\pm x\pm \sqrt{a^2-b^2}$
Hence, option 'C' is correct.