The area (in sq. units) of the region bounded by the parabola \(y^{2} = 12x\) and its latus rectum is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 24
Comparing with \(y^{2} = 4ax\):
\(4a = 12\), so \(a = 3\)
The latus rectum is the line x = a, that is, x = 3. The parabola is symmetric about the x-axis, so the area is twice the part above the x-axis, where \(y = \sqrt{12x} = 2\sqrt{3}\,\sqrt{x}\):
\( \displaystyle A = 2\int_{0}^{3} 2\sqrt{3}\,\sqrt{x}\,dx \)
\( \displaystyle = 4\sqrt{3}\left[\frac{2}{3}x^{3/2}\right]_{0}^{3} \)
\( \displaystyle = 4\sqrt{3}\times\frac{2}{3}\times 3\sqrt{3} \)
\(= 24\)