Question

The foci of an ellipse are $(0,\pm 4)$ and the equations for the directories are $y=\pm 9$ . The equation for the ellipse is

• A. $5x^2+9y^2=4$
• B. $2x^2-6y^2=28$
• C. $6x^2+3y^2=45$
• D. $9x^2+5y^2=180$

Answer 1

Answer: (D)

Given foci of ellipse are (0, - 4) and (0, 4). $\therefore$ Focal distance is 8. $\Rightarrow$ $2be=8$ $\Rightarrow$ $be=4$ ...(i) Also, since equation of directories are $y=\pm 9$ $\Rightarrow$ $\frac{b}{e}=9$ ...(ii) $\therefore$ From Eqs. (i) and (ii), we get $b^2=36$ $\Rightarrow$ $b=6$ and $e=\frac{2}{3}$ [from Eq. (i)] $\therefore$ $a^2=b^2(1-e^2)=36\left(1-\frac{4}{9}\right)=\frac{36\times 5}{9}=20$ So, equation of ellipse is $\frac{x^2}{20}+\frac{y^2}{36}=1$ $\Rightarrow$ $9x^2+5y^2=180$