Question

The slope(s) of the common tangent(s) to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ is/are

• A. $1$
• B. $2$
• C. $-1$
• D. $\frac{-1}{2}$

Answer 1

Answer: (A), (C)

$-1$

Let the slope of tangent is $m$, then
Tangent to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
$y=mx\pm \sqrt{a^2{m}^2-b^2}\cdots \left(1\right)$
The tangent to other hyperbola
$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ is
$y=mx\pm \sqrt{a^2-b^2{m}^2}\cdots \left(2\right)$
$\because$ both are same equations
$\therefore a^2{m}^2-b^2=a^2-b^2{m}^2$
$\Rightarrow (a^2+b^2)m^2=a^2+b^2$
$\Rightarrow m=\pm 1$