Question

The equation of the ellipse with foci at $(\pm 5,0)$ and $x=\frac{36}{5}$ as one directrix, is

• A. $\frac{x^2}{11}+\frac{y^2}{36}=1$
• B. $\frac{x^2}{36}+\frac{y^2}{11}=1$
• C. $\frac{x^2}{36}+\frac{y^2}{9}=1$
• D. $\frac{x^2}{9}+\frac{y^2}{36}=1$

Answer 1

The correct option is B $\frac{x^2}{36}+\frac{y^2}{11}=1$

$\because$ Focus is on $x-$ axis
Let the equation of ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$

Now foci of the ellipse will be $(\pm ae,0)$
$\therefore (\pm ae,0)\equiv (\pm 5,0)\Rightarrow ae=5\cdots \left(1\right)$
Equation of one of the directrices is
$x=\frac{a}{e}$ comparing it with given equation, we get
$\frac{a}{e}=\frac{36}{5}\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$, we get
$a^2=36\Rightarrow b^2=a^2-a^2{e}^2=11$

Hence, the equation of the ellipse is
$\frac{x^2}{36}+\frac{y^2}{11}=1(a>b)$