If `tan^(-1)(a/x)+tan^(-1)(b/x)=pi/2` then `x=`

Options

A. `ab`
B. `2ab`
C. `sqrtab`
D. `sqrt(2ab)`

Answer

To solve the equation \( \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) = \frac{\pi}{2} \), we can use a known identity involving inverse tangent functions.

### Step-by-Step Solution:

1. **Use the Identity**: We know that \( \tan^{-1}(u) + \tan^{-1}(v) = \frac{\pi}{2} \) if and only if \( uv = 1 \). In our case, let \( u = \frac{a}{x} \) and \( v = \frac{b}{x} \).

2. **Set Up the Equation**: According to the identity, we have:
\[ \frac{a}{x} \cdot \frac{b}{x} = 1 \]

3. **Simplify the Equation**: This simplifies to:
\[ \frac{ab}{x^2} = 1 \]

4. **Solve for \( x^2 \)**: Rearranging gives:
\[ ab = x^2 \]

5. **Take the Square Root**: Thus, we find:
\[ x = \sqrt{ab} \]

### Final Answer:
\[ x = \sqrt{ab} \]