Question

Find the equation of the ellipse whose focii are $(4,0)$ and $(-4,0)$ and eccentricity $e=\frac{1}{3}$.

• A. $\frac{x^2}{9}+\frac{y^2}{8}=\frac{16}{9}$
• B. $\frac{x^2}{8}+\frac{y^2}{9}=16$
• C. $\frac{x^2}{9}+\frac{y^2}{8}=32$
• D. None of these

Answer 1

Answer: (A)

$\frac{x^2}{9}+\frac{y^2}{8}=\frac{16}{9}$

$\Rightarrow a=4,e=\frac{1}{3}$

$\Rightarrow b^2=a^2(1-e^2)=16(1-\frac{1}{9})$

$\Rightarrow b^2=\frac{16\times 8}{9}\Rightarrow b=\frac{4}{3}\times 2\sqrt{2}$

Equation of ellipse : $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow \frac{x^2}{16}+\frac{9y^2}{128}=1$

$\Rightarrow \frac{x^2}{10\times 9}+\frac{y^2}{128}=\frac{1}{9}\Rightarrow \frac{x^2}{9}+\frac{y^2}{8}=\frac{16}{9}$