If `y=2sectheta and x=3tantheta` then `(x^(2))/(9)-(y^(2))/(4)`is

Options

A. 0
B. `-1`
C. 2
D. 1

Answer

To solve the problem, we need to evaluate the expression \(\frac{x^2}{9} - \frac{y^2}{4}\) given that \(y = 2 \sec \theta\) and \(x = 3 \tan \theta\).

### Step-by-Step Solution:

1. **Substitute the values of \(x\) and \(y\)**:
\[ x = 3 \tan \theta \quad \text{and} \quad y = 2 \sec \theta \]

2. **Calculate \(x^2\) and \(y^2\)**:
\[ x^2 = (3 \tan \theta)^2 = 9 \tan^2 \theta \]
\[ y^2 = (2 \sec \theta)^2 = 4 \sec^2 \theta \]

3. **Substitute \(x^2\) and \(y^2\) into the expression**:
\[ \frac{x^2}{9} - \frac{y^2}{4} = \frac{9 \tan^2 \theta}{9} - \frac{4 \sec^2 \theta}{4} \]

4. **Simplify the expression**:
\[ = \tan^2 \theta - \sec^2 \theta \]

5. **Use the identity \(\sec^2 \theta = 1 + \tan^2 \theta\)**:
\[ \tan^2 \theta - \sec^2 \theta = \tan^2 \theta - (1 + \tan^2 \theta) \]

6. **Combine like terms**:
\[ = \tan^2 \theta - 1 - \tan^2 \theta = -1 \]

Thus, the final result is:
\[ \frac{x^2}{9} - \frac{y^2}{4} = -1 \]

### Final Answer:
\(-1\)