Question

The equation of the 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

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

Explanation for the correct option:

Finding the equation of ellipse,

Given that distance between the foci is equal to $8$

$$\begin{gathered} \therefore 2ae=8 \\ ae=4 \end{gathered}$$

Distance between the directrix is $18$:

$$\begin{gathered} \therefore 2\left(\frac{a}{e}\right)=18 \\ \Rightarrow \frac{a}{e}=9 \end{gathered}$$

Equating the above equations, we get

$$\begin{gathered} ae=4 \\ \Rightarrow a.\left(\frac{a}{9}\right)=4\left[\text{Substituting}e=\frac{a}{9}\right] \\ \Rightarrow a=6 \\ \therefore e=\frac{4}{a}=\frac{4}{6}=\frac{2}{3} \end{gathered}$$

Calculating $b:$

$$\begin{gathered} b^2=a^2(1-e^2) \\ \Rightarrow b^2=a^2-{\left(ae\right)}^2 \\ \Rightarrow b^2=36-16 \\ \Rightarrow b^2=20 \end{gathered}$$

The equation of the ellipse is

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \Rightarrow \frac{x^2}{36}+\frac{y^2}{20}=1 \\ \Rightarrow 5x^2+9y^2=180 \end{gathered}$$

Hence, the correct answer is Option (D).