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

Options

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

Explanation

To solve the equation \( \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) = \frac{\pi}{2} \]

### Step 2: Use the Identity for Cotangent
Using the identity that states \( \tan^{-1}(u) + \tan^{-1}(v) = \frac{\pi}{2} \) implies \( uv = 1 \), we can rewrite our equation:
\[ \frac{a}{x} \cdot \frac{b}{x} = 1 \]

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

### Step 4: Solve for \( x^2 \)
Multiplying both sides by \( x^2 \) gives:
\[ ab = x^2 \]

### Step 5: Solve for \( x \)
Taking the square root of both sides, we find:
\[ x = \sqrt{ab} \]
Since \( x \) must be positive (as it represents a length or distance in this context), we take the positive root:
\[ x = \sqrt{ab} \]

### Final Answer
Thus, the solution is:
\[ \boxed{\sqrt{ab}} \]

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