What is the area of the parabola bounded by the latus rectum?

Question

Question

Direction : Consider the following for the items that follow :

The area bounded by the parabola y² = kx and the line x = k, where k > 0, is 4/3 square units.

This question was previously asked in

NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)

Answer 1

Explanation:

Given:

The area bounded by the parabola y² = kx and the line x = k, where k > 0, is 4/3 square units

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Area bounded = 4/3 sq. units

= \(2 \int_0^kydx\)

⇒ 4/3 = \(2 \int_0^k\sqrt k \sqrt xdx\)

⇒1 = k²

⇒ k = ±1 (since k>0)

⇒ k = 1

So equation of parabola is y² = x

⇒ 4a = 1

⇒ a - 1/4

Area of parabola y² = x and latus rectum at x = 1/4 is

= \(2 \int_0^\frac{1}{4}\sqrt x dx = 2\times \frac{2}{3}\times(\frac{1}{4})^\frac{3}{2}\)

= 1/6 sq. units

∴ Option (a) is correct

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