Question

The length of the latus rectum of the ellipse $16x^2+25y^2=400$ is

• A. $\frac{5}{6}$ units.
• B. $\frac{32}{5}$ units.
• C. $\frac{16}{5}$ units.
• D. $\frac{5}{32}$ units.

Answer 1

The correct option is B

$\frac{32}{5}$ units.

The explanation for the correct option

Given the equation of the ellipse: $16x^2+25y^2=400$.

Divide both sides of the equation by $400$.

$$\begin{gathered} \frac{16x^2}{400}+\frac{25y^2}{400}=\frac{400}{400} \\ \Rightarrow \frac{x^2}{25}+\frac{y^2}{16}=1 \\ \Rightarrow {\left(\frac{x}{5}\right)}^2+{\left(\frac{y}{4}\right)}^2=1 \end{gathered}$$

Compare the given equation with the standard equation of an ellipse, ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$.

Thus, $a=5,b=4$.

The length of the latus rectum can be given by $\frac{2b^2}{a}=\frac{2{\left(4\right)}^2}{5}\mathrm{units}=\frac{2\times 16}{5}\mathrm{units}=\frac{32}{5}\mathrm{units}$.

Therefore, the length of the latus rectum of the ellipse $16x^2+25y^2=400$ is $\frac{32}{5}$ units.