The equation of a directrix of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$ is

• A

  $3 \mathrm{y}=5$
• B

  $y=5$
• Correct answer
  C

  $3 y=25$
• D

  $y=3$

Solution

Equation of ellipse $\frac{\mathrm{x}^{2}}{\mathrm{~b}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1$, where $a>b$. Given, $\frac{x^{2}}{16}+\frac{y^{2}}{75}=1 \Rightarrow b=4, a=5$ But $\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\sqrt{1-\frac{16}{25}}$ $\Rightarrow \mathrm{e}=\frac{3}{5}$ $\therefore$ equation of directrix $\mathrm{y}=\pm \frac{\mathrm{a}}{\mathrm{e}}$ $\therefore \quad y=\pm \frac{5}{3 / 5} \Rightarrow 3 y=\pm 25$

Hence, the correct answer is option: 'C'.