Question

Area lying in the first quadrant and bounded by ellipse \(4x^2 + 9y^2 = 144\) is _____

• A. \( 24\pi \)
• B. \( 8\pi \)
• C. \( 12\pi \)
• D. \( 6\pi \)

Hint:

Area in first quadrant of symmetric curves like ellipse = one-fourth of total area.

Answer 1

The Correct Option is D

Concept: Standard equation of ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Area of full ellipse = \( \pi ab \)
Step 1: Convert to standard form. \[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \] So, \( a = 6, \; b = 4 \)
Step 2: Find total area. \[ \text{Area} = \pi ab = \pi \cdot 6 \cdot 4 = 24\pi \]
Step 3: First quadrant area. \[ \frac{1}{4} \times 24\pi = 6\pi \]