Question

What is the area bounded by the parabola $y^2=8x$ and its latus rectum.

• A. $\frac{16}{3}$sq units
• B. $\frac{32}{3}$sq units
• C. $\frac{8}{3}$sq units
• D. $\frac{64}{3}$sq units
• E. $\frac{4}{3}$sq units

Answer 1

The correct option is B

$\frac{32}{3}$sq units

Explanation of correct answer :

Finding the area bounded by the parabola and its latus rectum.

For parabola, $y^2=4ax$, the equation of latus rectum is $y=a$.

From the given equation of parabola, $y^2=8x$.

On comparing we get, $a=2$

therefore, Equation of latus rectum is, $y=2$.

The graph plotted for the curves is as shown:

$$\begin{gathered} y^2=8x \\ \Rightarrow y=\sqrt{8x} \end{gathered}$$

Required area $={\int }_0^2\sqrt{8x}dx$

$$\begin{gathered} =4\sqrt{2}{\left[\frac{x^{{\displaystyle \frac{3}{2}}}}{{\displaystyle \frac{3}{2}}}\right]}_0^2 \\ =\frac{32}{3}\text{sq units} \end{gathered}$$

Hence, the area within the parabola and its latus rectum is $\frac{32}{3}$sq units.

Thus, the correct answer is Option(B).