Question

Find the equation of the ellipse in the standard form given $e=\frac{1}{2}$ and it passes through $\left(2,1\right)$

Answer 1

Solution -

$\sqrt{1-\frac{b^2}{a^2}}=\frac{1}{2}$ $\frac{b^2}{a^2}=\frac{3}{4}$

$\frac{4}{a^2}+\frac{1}{b^2}=1$

$a^2=\frac{4b^2}{3}\to \frac{3}{6^2}+\frac{1}{6^2}=1$

$b^2=4$ $a^2=\frac{16}{3}$

Ellipse - $\frac{3x^2}{16}+\frac{y^2}{4}=1$