Question

The equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\pm 5),$ is

• A. $\frac{x^2}{10}+\frac{y^2}{5\sqrt{3}}=1$
• B. $\frac{x^2}{25}+\frac{y^2}{100}=1$
• C. $\frac{x^2}{75}+\frac{y^2}{100}=1$
• D. $\frac{x^2}{10}+\frac{y^2}{25}=1$

Answer 1

The correct option is C $\frac{x^2}{75}+\frac{y^2}{100}=1$

Since the foci are on $y-$axis, the major axis will be along the $y-$axis. So, equation of the ellipse is of the form
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b>a).$

Given,
Length of major axis $2b=20$
$\Rightarrow b=10$

and $be=5$
$\Rightarrow b^2{e}^2=25\Rightarrow b^2-a^2=25\Rightarrow a^2=75$

Hence, the equation of the ellipse is
$\frac{x^2}{75}+\frac{y^2}{100}=1.$