Question

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$

Answer 1

Let the ellipse equation be

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$e=\sqrt{\frac{2}{5}}$

$or,\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{\frac{2}{5}}$

Squaring both sides,

$5(a^2-b^2)=2a^2$

$or,a^2=\frac{5}{3}{b}^2$

And the ellipse passes through the point $(-3,1)$

$\frac{9}{a^2}+\frac{1}{b^2}=1$

$9b^2+a^2=a^2{b}^2$

$9b^2+\frac{5}{3}{b}^2=\frac{5}{3}{b}^4$

$\frac{32}{3}=\frac{5}{3}{b}^2$

$b^2=\frac{32}{5}$

$a^2=\frac{32}{3}$

Equation : $\frac{3x^2}{32}+\frac{5y^2}{32}=1$

$⟹3x^2+5y^2=32$