Question

If equation of the ellipse is $\frac{x^2}{100}+\frac{y^2}{400}=1$, then
I. Vertices of the ellipse are $(0,\pm 20)$
II. Foci of the ellipse are $(0,\pm 10\sqrt{3})$
III. Length of major axis is 40 .
IV. Eccentricity of the ellipse is $\frac{\sqrt{3}}{2}$.

• A. I and II are true.
• B. III and IV are true.
• C. II, III, IV are true.
• D. All are true.

Answer 1

Answer: (D)

$\frac{x^2}{100}+\frac{y^2}{400}=1$ is the equation of ellipse.
Major axis is along $y$ -axis
$a^2=400,\therefore a=20,b^2=100\therefore b=10$
$c^2=a^2-b^2=400-100=300\therefore c=10\sqrt{3}$
Vertices are $(0,\pm a)$ i.e., $(0,\pm 20)$
$\therefore$ Foci are $(0,\pm c)$ i.e., $(0,\pm 10\sqrt{3})$
Length of major axis $=2a=2\times 20=40$
Length of minor axis $=2b=2\times 10=20$
Eccentricity, $e=\frac{c}{a}=\frac{10\sqrt{3}}{20}=\frac{\sqrt{3}}{2}$
Length $=\frac{2b^2}{a}=\frac{2\times 100}{20}=10$