If `x = 2 (t +(1)/(t)) and y = 3 (t - (1)/( t )),` then ` (x ^(2))/( 4 ) - (y ^(2))/( 9)` is

Options

A. 3
B. `-4`
C. `4`
D. `-3`

Answer

To solve the problem, we need to find the value of the expression \( \frac{x^2}{4} - \frac{y^2}{9} \) given that \( x = 2 \left( t + \frac{1}{t} \right) \) and \( y = 3 \left( t - \frac{1}{t} \right) \).

### Step-by-step Solution:

1. **Substitute the values of \(x\) and \(y\)**:
\[ x = 2 \left( t + \frac{1}{t} \right) \quad \text{and} \quad y = 3 \left( t - \frac{1}{t} \right) \]

2. **Calculate \(x^2\)**:
\[ x^2 = \left( 2 \left( t + \frac{1}{t} \right) \right)^2 = 4 \left( t + \frac{1}{t} \right)^2 \]

3. **Expand \( (t + \frac{1}{t})^2 \)**:
\[ (t + \frac{1}{t})^2 = t^2 + 2 + \frac{1}{t^2} \]
Therefore,
\[ x^2 = 4 \left( t^2 + 2 + \frac{1}{t^2} \right) = 4t^2 + 8 + \frac{4}{t^2} \]

4. **Calculate \(y^2\)**:
\[ y^2 = \left( 3 \left( t - \frac{1}{t} \right) \right)^2 = 9 \left( t - \frac{1}{t} \right)^2 \]

5. **Expand \( (t - \frac{1}{t})^2 \)**:
\[ (t - \frac{1}{t})^2 = t^2 - 2 + \frac{1}{t^2} \]
Therefore,
\[ y^2 = 9 \left( t^2 - 2 + \frac{1}{t^2} \right) = 9t^2 - 18 + \frac{9}{t^2} \]

6. **Substitute \(x^2\) and \(y^2\) into the expression**:
\[ \frac{x^2}{4} - \frac{y^2}{9} = \frac{4t^2 + 8 + \frac{4}{t^2}}{4} - \frac{9t^2 - 18 + \frac{9}{t^2}}{9} \]

7. **Simplify \(\frac{x^2}{4}\)**:
\[ \frac{x^2}{4} = t^2 + 2 + \frac{1}{t^2} \]

8. **Simplify \(\frac{y^2}{9}\)**:
\[ \frac{y^2}{9} = t^2 - 2 + \frac{1}{t^2} \]

9. **Combine the two simplified expressions**:
\[ \frac{x^2}{4} - \frac{y^2}{9} = \left( t^2 + 2 + \frac{1}{t^2} \right) - \left( t^2 - 2 + \frac{1}{t^2} \right) \]

10. **Simplify the expression**:
\[ = t^2 + 2 + \frac{1}{t^2} - t^2 + 2 - \frac{1}{t^2} \]
\[ = 4 \]

### Final Answer:
The value of \( \frac{x^2}{4} - \frac{y^2}{9} \) is \( 4 \).