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 $=2 a=2 \times 20=40$
Length of minor axis $=2 b=2 \times 10=20$
Eccentricity, $e=\frac{c}{a}=\frac{10 \sqrt{3}}{20}=\frac{\sqrt{3}}{2}$
Length $=\frac{2 b^{2}}{a}=\frac{2 \times 100}{20}=10$