Q100 / 180 · COMEDK 2024 (Evening Shift) · Mathematics
$\text { The area of the region enclosed by the curve }\left\{(x, y): 4 x^2+25 y^2=100\right\} \text { is }$
- A$9 \pi$ sq units
- Correct answerB$10 \pi$ sq units
- C$\dfrac{9 \pi}{5} \text { sq units }$
- D$\dfrac{16 \pi}{3} \text { sq units }$
Solution
The given equation is $4x^2 + 25y^2 = 100$.
Dividing both sides by $100$, we get $\dfrac{4x^2}{100} + \dfrac{25y^2}{100} = 1$, which simplifies to $\dfrac{x^2}{25} + \dfrac{y^2}{4} = 1$.
This is the standard equation of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, where $a^2 = 25$ and $b^2 = 4$.
Thus, $a = 5$ and $b = 2$.
The area of an ellipse is given by the formula $A = \pi ab$.
Substituting the values of $a$ and $b$, we get $A = \pi \times 5 \times 2 = 10\pi$ sq units.
Answer: $10 \pi$ sq units
Dividing both sides by $100$, we get $\dfrac{4x^2}{100} + \dfrac{25y^2}{100} = 1$, which simplifies to $\dfrac{x^2}{25} + \dfrac{y^2}{4} = 1$.
This is the standard equation of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, where $a^2 = 25$ and $b^2 = 4$.
Thus, $a = 5$ and $b = 2$.
The area of an ellipse is given by the formula $A = \pi ab$.
Substituting the values of $a$ and $b$, we get $A = \pi \times 5 \times 2 = 10\pi$ sq units.
Answer: $10 \pi$ sq units
Hence, the correct answer is option: 'B'.