Question

Find the area of the region bounded by the parabola $$ y^2 = 4ax \text{ and its latus rectum.} $$ 

• A. \( \frac{a^2}{2} \)
• B. \( a^2 \)
• C. \( \frac{a^2}{4} \)
• D. \( \frac{a^2}{8} \)

Hint:

To find the area under a curve, integrate the function over the desired limits. For curves like parabolas, carefully handle the limits and the integrals.

Answer 1

The Correct Option is A

Step 1: The equation of the parabola is given by: \[ y^2 = 4ax \] The latus rectum of the parabola is the line parallel to the directrix passing through the focus. The focus is at \( (a, 0) \), and the latus rectum is the line \( x = a \). 

Step 2: To find the area, we need to integrate the area between the parabola and the line \( x = a \). The parabola equation gives \( y = \pm \sqrt{4ax} \). The area bounded by the parabola and the latus rectum can be calculated as: \[ \text{Area} = 2 \int_0^a \sqrt{4ax} \, dx \] 

Step 3: Simplify the integral: \[ \text{Area} = 2 \int_0^a \sqrt{4a} \sqrt{x} \, dx = 2 \sqrt{4a} \int_0^a \sqrt{x} \, dx \] \[ = 2 \sqrt{4a} \cdot \left[ \frac{2}{3} x^{3/2} \right]_0^a = 2 \sqrt{4a} \cdot \frac{2}{3} a^{3/2} \] \[ = \frac{4}{3} a^{2} \] Thus, the area is \( \frac{a^2}{2} \).