Question

The area of the region enclosed by the curve $ \left\{ (x, y) : 4x^2 + 25y^2 = 100 \right\} $ is:

• A. \( \frac{16\pi}{3} \) sq units
• B. \( 9\pi \) sq units
• C. \( 10\pi \) sq units
• D. \( \frac{9\pi}{5} \) sq units

Hint:

To find the area of an ellipse, use the formula \( A = \pi \cdot a \cdot b \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.

Answer 1

The Correct Option is C

The given equation represents an ellipse in standard form, which can be written as: \[ \frac{x^2}{25} + \frac{y^2}{4} = 1. \] For an ellipse, the area \( A \) is given by the formula: \[ A = \pi \cdot a \cdot b, \] where \( a \) and \( b \) are the semi-major and semi-minor axes of the ellipse. In this case, \( a = 5 \) and \( b = 2 \) (since the equation of the ellipse is \( \frac{x^2}{5^2} + \frac{y^2}{2^2} = 1 \)). Substituting these values into the area formula: \[ A = \pi \cdot 5 \cdot 2 = 10\pi \text{ sq units}. \] Thus, the area of the region enclosed by the curve is \( 10\pi \) sq units.