Question

The area of the region bounded by the parabola \[ y^2 = 8x \quad \text{and its latus rectum is} \]

• A. \( \frac{16}{3} \) sq. units
• B. \( \frac{8}{3} \) sq. units
• C. \( \frac{32}{3} \) sq. units
• D. \( \frac{4}{3} \) sq. units

Hint:

For a parabola \( y^2 = 4ax \), the area between the parabola and its latus rectum can be calculated using integration and the length of the latus rectum.

Answer 1

The Correct Option is C

Step 1: Equation of the parabola.
The equation of the parabola is \( y^2 = 8x \). The latus rectum of a parabola is the line segment passing through the focus and perpendicular to the axis of symmetry. For a parabola of the form \( y^2 = 4ax \), the length of the latus rectum is \( 4a \). For \( y^2 = 8x \), we have \( 4a = 8 \), so the length of the latus rectum is 8 units.
Step 2: Find the area.
The area of the region bounded by the parabola and its latus rectum can be calculated by integrating the equation of the parabola from 0 to 8. The equation of the parabola gives \( y = \pm \sqrt{8x} \). Therefore, the area is given by: \[ A = 2 \int_0^8 \sqrt{8x} \, dx. \] Simplifying the integral, we get: \[ A = 2 \times \left[ \frac{2}{3} \times 8^{3/2} \right] = \frac{32}{3}. \]
Step 3: Conclusion.
Thus, the area of the region is \( \frac{32}{3} \) sq. units, which corresponds to option (C).