Question

The area bounded by the parabola \( y^2 = 16x \) and its latus-rectum in the first quadrant is

• A. \( 128 \, \text{sq. units} \)
• B. \( \frac{64}{3} \, \text{sq. units} \)
• C. \( \frac{128}{3} \, \text{sq. units} \)
• D. \( 64 \, \text{sq. units} \)

Hint:

When calculating the area under a curve, set up the integral based on the equation of the curve and the boundaries of the region.

Answer 1

The Correct Option is B

Step 1: Find the equation of the latus rectum.
The equation of the latus rectum of the parabola \( y^2 = 16x \) is \( x = 4 \), and the length of the latus rectum is \( 4 \).
Step 2: Calculate the area.
The area bounded by the parabola and the latus rectum in the first quadrant is given by: \[ A = \int_0^4 \sqrt{16x} \, dx = \int_0^4 4\sqrt{x} \, dx \] Evaluating this integral gives: \[ A = \frac{64}{3} \, \text{sq. units} \] Step 3: Conclusion.
The correct answer is (B) \( \frac{64}{3} \, \text{sq. units} \).