Question

The equation of ellipse whose distance between the foci is equal to $8$ and distance between the directrix is $18,$ is -

• 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 $5x^2+9y^2=180$

$\Rightarrow$ Distance between two foci$=ae-(-ae)$

$=2ae=8$ ………….$\left(1\right)$

$\Rightarrow$ Distance between two directrix$=\frac{a}{e}-\left(\frac{-a}{2}\right)$

$=\frac{2a}{e}=18$ ………..$\left(2\right)$

Multiply equation $\left(1\right)$ & equation $\left(2\right)$

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

$4a^2=144$

$a^2=36$

$\Rightarrow a=6$

From equation $\left(1\right)$

$2ae=8$

$\Rightarrow e=\frac{2}{3}$

From the formula,

$b^2=a^2(1-e^2)$

$\Rightarrow 6^2(1-{\left(\frac{2}{3}\right)}^2)$

$=36(1-\frac{4}{9})$

$=36\left(\frac{5}{9}\right)=20$

Equation of the ellipse is:-

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

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

$\Rightarrow \frac{5x^2+9y^2}{100}=1$

$\Rightarrow 5x^2+9y^2=100$.