Question

The area of the region \[\left\{ (x, y) : y^2 \leq 4x, \, x<4, \, \frac{xy(x - 1)(x - 2)}{(x - 3)(x - 4)}>0, \, x \neq 3 \right\}\]is

• A. \( \frac{16}{3} \)
• B. \( \frac{64}{3} \)
• C. \( \frac{8}{3} \)
• D. \( \frac{32}{3} \)

Answer 1

The Correct Option is D

\[ y^2 \leq 4x, \quad x < 4 \]

\[ \frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0 \]

Case - I: \( y > 0 \)

\[ \frac{x(x-1)(x-2)}{(x-3)(x-4)} > 0, \quad x \in (0,1) \cup (2,3) \]

Case - II: \( y < 0 \)

\[ \frac{x(x-1)(x-2)}{(x-3)(x-4)} < 0, \quad x \in (1,2) \cup (3,4) \]

Area:

\[ \text{Area} = 2 \int_{0}^{4} \sqrt{x} \, dx \]

\[ = 2 \cdot \frac{2}{3} \left[ x^{3/2} \right]_{0}^{4} = \frac{32}{3} \]