Question

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

• A. $3y=\pm 5$
• B. $y=\pm 5$
• C. $3y=\pm 25$
• D. $y=\pm 3$

Answer 1

The correct option is C

$3y=\pm 25$

Explanation for the correct option:

The general equation of ellipse is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ , where $a>b$.

Given, $$\begin{gathered} \frac{x^2}{16}+\frac{y^2}{25}=1 \\ \Rightarrow b=4,a=5 \end{gathered}$$

Step 1: Calculating the value of $e$,

$\begin{array}{rcl}e& =& \sqrt{1-\frac{b^2}{a^2}}\\ & =& \sqrt{1-\frac{{16}^2}{{25}^2}}\\ & =& \frac{3}{5}\end{array}$

Equation of directrix, $y=\pm \frac{a}{e}$

Step 2 :Substituting the values of $a\text{and}e$. To Calculate the value of $y:$

$$\begin{gathered} y=\pm \frac{5}{{\displaystyle \frac{3}{5}}} \\ \Rightarrow 3y=\pm 25 \end{gathered}$$

Hence, option (C) is the correct answer.