What is the value of k?
Direction: Consider the following for the two (02) items that follow : The area bounded by the parabola y^{2}=kx and the line x=k, where k \gt 0, is \frac{4}{3} square units.
- A. 1/2
- B. 1 ✓
- C. \sqrt{2}
- D. 2
Correct Answer: B. 1
Explanation
The parabola is symmetric about the x-axis. The area is given by A = 2\int_{0}^{k} \sqrt{kx} \,dx = 2\sqrt{k} \left[\frac{2}{3}x^{3/2}\right]_{0}^{k} = \frac{4}{3}\sqrt{k}\cdot k^{3/2} = \frac{4}{3}k^{2}. Equating this to \frac{4}{3} gives \frac{4}{3}k^2 = \frac{4}{3} \implies k^2 = 1. Since k \gt 0, k = 1.
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