If `ax=(2pq)/(1+q^(2))`, find the value of `(sqrt((p//a)+x)+sqrt((p//a)-x))/(sqrt([(p//a)+x])-sqrt([(p//a)-x]))`

Answer

To solve the problem, we start with the given equation:

\[ ax = \frac{2pq}{1 + q^2} \]

We need to find the value of the expression:

\[ \frac{\sqrt{\left(\frac{p}{a}\right) + x} + \sqrt{\left(\frac{p}{a}\right) - x}}{\sqrt{\left(\frac{p}{a}\right) + x} - \sqrt{\left(\frac{p}{a}\right) - x}} \]

### Step 1: Express \(\frac{p}{a}\) in terms of \(x\)

From the equation \(ax = \frac{2pq}{1 + q^2}\), we can isolate \(\frac{p}{a}\):

\[ \frac{p}{a} = \frac{ax(1 + q^2)}{2q} \]

### Step 2: Substitute \(\frac{p}{a}\) into the expression

Now, we substitute \(\frac{p}{a}\) into the expression we need to evaluate:

\[ \frac{\sqrt{\left(\frac{ax(1 + q^2)}{2q}\right) + x} + \sqrt{\left(\frac{ax(1 + q^2)}{2q}\right) - x}}{\sqrt{\left(\frac{ax(1 + q^2)}{2q}\right) + x} - \sqrt{\left(\frac{ax(1 + q^2)}{2q}\right) - x}} \]

### Step 3: Simplify the expression

Let \(y = \frac{p}{a}\). Then we rewrite the expression as:

\[ \frac{\sqrt{y + x} + \sqrt{y - x}}{\sqrt{y + x} - \sqrt{y - x}} \]

### Step 4: Use the identity for the sum and difference of square roots

We can use the identity:

\[ \sqrt{a} + \sqrt{b} = \frac{(\sqrt{a} + \sqrt{b})^2}{\sqrt{a} - \sqrt{b}} \]

Thus, we can rewrite the expression as:

\[ \frac{(\sqrt{y + x} + \sqrt{y - x})^2}{(\sqrt{y + x} - \sqrt{y - x})^2} \]

### Step 5: Calculate the numerator and denominator

The numerator becomes:

\[ (y + x) + (y - x) + 2\sqrt{(y + x)(y - x)} = 2y + 2\sqrt{y^2 - x^2} \]

The denominator becomes:

\[ (y + x) - (y - x) = 2x \]

### Step 6: Final simplification

Thus, we have:

\[ \frac{2y + 2\sqrt{y^2 - x^2}}{2x} = \frac{y + \sqrt{y^2 - x^2}}{x} \]

### Step 7: Substitute back for \(y\)

Substituting back \(y = \frac{p}{a}\):

\[ \frac{\frac{p}{a} + \sqrt{\left(\frac{p}{a}\right)^2 - x^2}}{x} \]

### Step 8: Final result

The final result simplifies to:

\[ \frac{1}{q} \]