Question

For the given ellipse, find the length of major and minor axes, coordinates of foci and vertices and the eccentricity.

$16x^2+25y^2=400$

Answer 1

For an ellipse of the form:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b>0$)

Length of major axis: $2a$

Length of minor axis: $2b$

Eccentricity (e): $\sqrt{1-\frac{b^2}{a^2}}$

Foci: $(\pm ae,0)$

Vertices: $(\pm a,0)$
Given ellipse:

$16x^2+25y^2=400⟹\frac{x^2}{25}+\frac{y^2}{16}=1$

Therefore, $a=5,b=4$
Now,

Length of major axis: $2a=10$

Length of minor axis: $2b=8$

Eccentricity (e): $\sqrt{1-\frac{b^2}{a^2}}=\frac{3}{5}$

Foci: $(\pm 3,0)$

Vertices: $(\pm 5,0)$