Q210 / 270 · NDA 2022 (Phase 1) · Mathematics

The area of the region bounded by the parabola $y^2=4 k x$, where $k \gt 0$ and its latus rectum is 24 square units. What is the value of $k$ ?

  • A
    1
  • B
    2
  • Correct answer
    C
    3
  • D
    4

Solution

Required area $\begin{aligned} & \text { 24. }=2 \int_0^{\mathrm{k}} 2 \sqrt{\mathrm{k}} \sqrt{\mathrm{x}} \mathrm{dx} \\ & \Rightarrow 2 \sqrt{\mathrm{k}}\left[\frac{2}{3} \mathrm{x}^{\frac{3}{2}}\right]_0^{\mathrm{k}}=12 \\ & \Rightarrow \mathrm{k}^{\frac{1}{2}} \times \frac{2}{3} \cdot \mathrm{k}^{\frac{3}{2}}=6 \\ & \Rightarrow \mathrm{k}^2=6 \times \frac{3}{2}=9 \\ & \Rightarrow \mathrm{k}=3(\because \mathrm{k} \gt 0)\end{aligned}$

Hence, the correct answer is option: 'C'.

Relevant Questions