Question

The area bounded by the parabola $y^{2 = 4ax$ and its latus rectum is:

• A. $\frac{8}{3} a^{2}$
• B. $\frac{4}{3} a^{2}$
• C. $\frac{2}{3} a^{2}$
• D. $\frac{16}{3} a^{2}$

Hint:

This area is exactly $2/3$ of the area of the rectangle formed by the latus rectum and the vertex.

Answer 1

The Correct Option is A

Step 1: Concept
The latus rectum of $y^{2} = 4ax$ is the line $x = a$.
Step 2: Analysis
Area $= 2 \int_{0}^{a} y dx = 2 \int_{0}^{a} \sqrt{4ax} dx = 4\sqrt{a} \int_{0}^{a} x^{1/2} dx$. $= 4\sqrt{a} [\frac{2}{3} x^{3/2}]_{0}^{a} = \frac{8}{3} \sqrt{a} \cdot a^{3/2}$.
Step 3: Conclusion
Area $= \frac{8}{3} a^{2}$.
Final Answer: (A)