Question

The equation of ellipse whose distance between the foci is equal to $8$ and the distance between the directrix is $18$ is $(a>b)$-

• A. $5x^2-9y^2=180$
• B. $9x^2+5y^2=180$
• C. $x^2+9y^2=180$
• D. $5x^2+9y^2=180$

Answer 1

The correct option is D $5x^2+9y^2=180$

$ae-(-ae)$

$2ae-8$

$\frac{a}{e}-\left(\frac{-a}{e}\right)$

$\frac{2a}{e}=18$

$22ae\times \frac{2a}{e}=8\times 18$

$4a^2=18\times 8$

$a^2=36$ $a=\pm 6$

$2ae=8$

$2\left(6\right)e=8\Rightarrow e=\frac{2}{3}$

$e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{b^2}{36}}$

${\left(\frac{2}{3}\right)}^2=1-\frac{b^2}{36}$

$\frac{b^2}{36}=1-\frac{4}{9}=\frac{5}{9}$

$b^2=20$

$\frac{x^2}{36}+\frac{y^2}{20}=1$

$5x^2+9y^2=180$