Question

The area cut off by the latus rectum from the parabola $y^2=4ax$ is

• A. $\frac{8}{3}a$
• B. $\frac{8}{3}\sqrt{a}$
• C. $\frac{3}{8}{a}^2$
• D. $\frac{8}{3}{a}^2$

Answer 1

The correct option is D

$\frac{8}{3}{a}^2$

Explanation for correct option

Finding the area cut off by the latus rectum off the parabola

Latus rectum is the line $x=a$

Therefore, the required area will be,

$\begin{array}{rcl}& =& AreaOLSL\text{'}\\ & =& 2\times AreaOSL\\ & =& 2\times {\int }_0^{a}ydx\\ & =& 2\times {\int }_0^a\sqrt{4ax}dx(\because y^2=4ax)\Rightarrow y=4ax)\\ & =& 2\times {\int }_0^a\sqrt{4a}\sqrt{x}dx\\ & =& 4\sqrt{a}{\int }_0^a\sqrt{x}dx\\ & =& 4\sqrt{a}{\int }_0^a\sqrt{x}dx\\ & =& \frac{8}{3}\sqrt{a}{a}^{\frac{3}{2}}\\ & =& \frac{8}{3}{a}^{\frac{4}{2}}\\ & =& \frac{8}{3}{a}^2\end{array}$

Hence, the correct answer is Option (D).