Question

Which of following is incorrect regarding eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{and}{c}^2=a^2+b^2?$

• A. $e=\sqrt{1+\frac{a^2}{b^2}}$
• B. $e=\sqrt{1+\frac{b^2}{a^2}}$
• C. $e=\frac{\text{Distance from centre to focus}}{\text{Distance from centre to Vertex}}$
• D. $e=\frac{c}{a}$

Answer 1

The correct option is A $e=\sqrt{1+\frac{a^2}{b^2}}$

As we know,
$e=\frac{\text{Distance from centre to focus}}{\text{Distance from centre to Vertex}}$
For a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,
$e=\frac{c}{a}\text{and}{c}^2=a^2+b^2$
$\Rightarrow a^2{e}^2=a^2+b^2$
$\Rightarrow e^2=\frac{a^2+b^2}{a^2}$
$\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}$