Question

The equations of common tangents to the 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. $x+y=\sqrt{a^2-b^2}$
• B. $x-y=\sqrt{a^2-b^2}$
• C. $x-y=-\sqrt{a^2+b^2}$
• D. $x-y=-\sqrt{a^2-b^2}$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $x+y=\sqrt{a^2-b^2}$
B $x-y=\sqrt{a^2-b^2}$
D $x-y=-\sqrt{a^2-b^2}$

The curves are mirror images of each other in the two lines, $x-y=0$ and $x+y=0.$ The slope of the common tangent, is $m=\pm 1$.

Since, equation of a tangent of slope $m$ to a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $y=mx\pm \sqrt{a^2-b^2}$

$\therefore$ Equation of the common tangents are $y=\pm x\pm \sqrt{a^2-b^2}$