Statement-I: The area bounded by `x=2 cos theta,y=3sin theta` is `36pi` sq. units.
Statement-II : The area bounded by `x=2cos theta, y=2 sin theta` is `4pi` sq. units.
Which of the above statement is correct.

Options

A. only I
B. only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze both statements regarding the areas bounded by the given curves.

### Step 1: Analyze Statement I
The first statement claims that the area bounded by the curves \( x = 2 \cos \theta \) and \( y = 3 \sin \theta \) is \( 36\pi \) square units.

1. **Convert Parametric Equations to Cartesian Form**:
- We have \( x = 2 \cos \theta \) and \( y = 3 \sin \theta \).
- Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \left(\frac{y}{3}\right)^2 + \left(\frac{x}{2}\right)^2 = 1 \]
- This represents an ellipse with semi-major axis \( b = 3 \) and semi-minor axis \( a = 2 \).

2. **Calculate the Area of the Ellipse**:
- The area \( A \) of an ellipse is given by the formula:
\[ A = \pi \times a \times b \]
- Substituting the values:
\[ A = \pi \times 2 \times 3 = 6\pi \]
- Therefore, the area bounded by the first statement is \( 6\pi \) square units, not \( 36\pi \).

### Conclusion for Statement I:
- **Statement I is false** because the calculated area is \( 6\pi \) square units.

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### Step 2: Analyze Statement II
The second statement claims that the area bounded by the curves \( x = 2 \cos \theta \) and \( y = 2 \sin \theta \) is \( 4\pi \) square units.

1. **Convert Parametric Equations to Cartesian Form**:
- For \( x = 2 \cos \theta \) and \( y = 2 \sin \theta \):
\[ \left(\frac{y}{2}\right)^2 + \left(\frac{x}{2}\right)^2 = 1 \]
- This represents a circle with radius \( r = 2 \).

2. **Calculate the Area of the Circle**:
- The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
- Substituting the radius:
\[ A = \pi \times (2)^2 = 4\pi \]
- Therefore, the area bounded by the second statement is \( 4\pi \) square units.

### Conclusion for Statement II:
- **Statement II is true** because the calculated area is indeed \( 4\pi \) square units.

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### Final Conclusion:
- Statement I is false.
- Statement II is true.

### Summary:
- **Correct Statement**: Only Statement II is correct.

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