Question

If the semi-major axis of an ellipse is 3 and the latus rectum 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. $\frac{3x^{2}}{8}+\frac{y^{2}}{9}=1$

Answer 1

Answer: (C)

$\frac{x^{2}}{9}+\frac{3y^{2}}{8}=1$

Answer 2

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 $=a=3$
Latusrectum $=\frac{2 b^{2}}{a}=\frac{16}{9}$
$\Rightarrow b^{2}=\frac{8}{9} a=\frac{8}{9} \times 3=\frac{8}{3}$
$\therefore$ Equation of ellipse is
$\frac{x^{2}}{9}+\frac{3 y^{2}}{8}=1$