Question

Find equation ellipse whose centre lies at the origin, major lies on $X-$axis, whose eccentricity is $\frac{2}{3}$ and whose length of latus rectum is $5$ units.

Answer 1

general equation as ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$e=\frac{2}{3}$ and latus ractus $=5$

Latus ractus $=\frac{2b^2}{a}=5$

$=b^2=\frac{5a}{2}$...(1)

Also $b^2=a^2(a-l^2)$

$\frac{59}{2}=a^2(1-\frac{4}{5})$

$\frac{59}{2}=a^2\times \frac{5}{9}$

$a=\frac{9}{2}\Rightarrow b^2=\frac{5\times 9}{2}$

$a^2=\frac{81}{4}$ $b^2=\frac{45}{2}$

Equation

$\frac{x^2}{\left(\frac{81}{4}\right)}+\frac{y^2}{\left(\frac{45}{2}\right)}=1$

$20x^2+36y^2=405$.