Question

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

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

Answer 1

The correct option is B

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

Let y =mx+c be a common tangent of hyperbolas
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\mathrm{1............}\left(i\right)and\frac{y^2}{a^2}-\frac{x^2}{b^2}=\mathrm{1............}\left(ii\right)$
Condition of tangency for Eq. (i) is
$c^2=a^2{m}^2-b^2................\left(iii\right)$
and condition of tangency for Eq. (ii) is
$c^2=a^2-b^2{m}^2.........................\left(iv\right)$
From Eqs. (iii) and (iv),
$a^2{m}^2-b^2=a^2-b^2{m}^2\Rightarrow a^2(m^2-1)+b^2(m^2-1)=0\Rightarrow (a^2+b^2)(m^2-1)=0\because a^2+b^2\ne 0\therefore m^2-1=0\Rightarrow m=\pm 1FromEq.\left(iii\right),c^2=a^2-b^{2}c=\pm \sqrt{(a^2-b^2)}$
Hence, equations of common tangents are
$y=\pm x\pm \sqrt{(a^2-b^2)}$