Question

What is the eccentricity of a rectangular hyperbola?

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $\sqrt{5}$
• D. $\sqrt{6}$

Answer 1

The correct option is A

$\sqrt{2}$

Explanation for the correct option:

The general equation of a hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a$ represents the transverse of hyperbola and $b$ represents the conjugate of the hyperbola.

The eccentricity of a hyperbola is $e=\frac{\sqrt{a^2+b^2}}{a}\cdots \left(1\right)$.

Since a rectangular hyperbola is defined as a hyperbola whose transverse axis is identical to its conjugate axis, $a=b$.

Substitute $a=b$into $\left(1\right)$.

$\begin{array}{rcl}e& =& \frac{\sqrt{a^2+b^2}}{a}\\ & \Rightarrow & e=\frac{\sqrt{a^2+a^2}}{a}\\ & \Rightarrow & e=\frac{\sqrt{2a^2}}{a}\\ & \Rightarrow & e=\frac{\sqrt{2}a}{a}\\ & \Rightarrow & e=\sqrt{2}\end{array}$

Hence, option (A) is correct.