Question

If the length of the major axis of an ellipse is $\frac{17}{8}$ times the length of the minor axis, then the eccentricity of the ellipse is

• A. $\frac{8}{17}$
• B. $\frac{15}{17}$
• C. $\frac{9}{17}$
• D. $\frac{2\sqrt{2}}{17}$
• E. $\frac{13}{17}$

Answer 1

The correct option is B

$\frac{15}{17}$

Explanation for the correct option:

Let us assume the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\left(a>b\right)$.

The length of the major axis can be given by $2a$.

The length of the major axis can be given by $2b$.

Thus, $2a=\frac{17}{8}\times 2b$

$\Rightarrow \frac{b}{a}=\frac{8}{17}$.

Therefore the eccentricity of the ellipse can be given by, $e=\sqrt{1-{\left(\frac{b}{a}\right)}^2}$

Put $\frac{b}{a}=\frac{8}{17}$.

$$\begin{gathered} \Rightarrow e=\sqrt{1-{\left(\frac{8}{17}\right)}^2} \\ \Rightarrow e=\sqrt{1-\frac{64}{289}} \\ \Rightarrow e=\sqrt{\frac{289-64}{289}} \\ \Rightarrow e=\sqrt{\frac{225}{289}} \\ \Rightarrow e=\frac{15}{17} \end{gathered}$$

Hence, option (B) is the correct answer.