Question

The eccentricity of the conic $\left[\frac{{\left(\mathrm{x}+2\right)}^2}{7}\right]+{\left(\mathrm{y}-1\right)}^2=14$ is

• A. $\sqrt{\frac{7}{8}}$
• B. $\sqrt{\frac{6}{17}}$
• C. $\sqrt{\frac{3}{2}}$
• D. $\sqrt{\frac{6}{11}}$
• E. $\sqrt{\frac{6}{7}}$

Answer 1

The correct option is E

$\sqrt{\frac{6}{7}}$

Explanation for correct option:

Given the equation of the conic is $\left[\frac{{\left(\mathrm{x}+2\right)}^2}{7}\right]+{\left(\mathrm{y}-1\right)}^2=14$

The equation of the ellipse becomes $\left[\frac{{\left(\mathrm{x}+2\right)}^2}{7\times 14}\right]+\frac{{\left(\mathrm{y}-1\right)}^2}{14}=1$

$$\begin{gathered} \Rightarrow a^2=7\times 14 \\ \Rightarrow b^2=14 \\ \Rightarrow \mathit{e}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{a}}^{\mathbf{2}}}} \\ =\sqrt{1-\frac{14}{7\times 14}} \\ =\sqrt{1-\frac{1}{7}} \\ =\sqrt{\frac{6}{7}} \end{gathered}$$

Hence, Option $\left(\mathrm{E}\right)$ is the correct answer