Question

If the semi-major axis of an ellipse is $3$ and the latusrectum is $\frac{16}{9}$, then the standard equation of the ellipse is

• A. $\frac{x^2}{9}+\frac{y^2}{8}=1$
• B. $\frac{x^2}{8}+\frac{y^2}{9}=1$
• C. $\frac{x^2}{9}+\frac{3y^2}{8}=1$
• D. $\frac{3x^2}{8}+\frac{y^2}{9}=1$
• E. $\frac{x^2}{9}+\frac{8y^2}{3}=1$

Answer 1

The correct option is C $\frac{x^2}{9}+\frac{3y^2}{8}=1$

Let the equation of ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$
Now, it is given that semi-major axis $=a=3$
Latus rectum $=\frac{2b^2}{a}=\frac{16}{9}$
$\Rightarrow b^2=\frac{8}{9}a=\frac{8}{9}\times 3=\frac{8}{3}$
Therefore, equation of ellipse is
$\frac{x^2}{9}+\frac{3y^2}{8}=1$