If `( 3 x - 2y) : ( 2x + 3y) = 5 : 6 ` then one of the value of `((3 sqrt( x) + 3 sqrt( y))/(3 sqrt(x) - 3 sqrt ( y)))^(2)` is

Options

A. `(1)/(5)`
B. 5
C. 25
D. `(1)/(25)`

Answer

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

\[ \frac{3x - 2y}{2x + 3y} = \frac{5}{6} \]

### Step 1: Cross Multiply
Cross multiplying gives us:

\[ 6(3x - 2y) = 5(2x + 3y) \]

### Step 2: Expand Both Sides
Expanding both sides:

\[ 18x - 12y = 10x + 15y \]

### Step 3: Rearranging the Equation
Now, we rearrange the equation to group the terms involving \(x\) and \(y\):

\[ 18x - 10x = 15y + 12y \]

This simplifies to:

\[ 8x = 27y \]

### Step 4: Express \(x\) in terms of \(y\)
From the equation \(8x = 27y\), we can express \(x\) in terms of \(y\):

\[ x = \frac{27}{8}y \]

### Step 5: Find the Ratio of \(\sqrt{x}\) and \(\sqrt{y}\)
Now we need to find the ratio of \(\sqrt{x}\) and \(\sqrt{y}\):

\[ \sqrt{x} = \sqrt{\frac{27}{8}y} = \frac{\sqrt{27}}{\sqrt{8}} \sqrt{y} = \frac{3\sqrt{3}}{2\sqrt{2}} \sqrt{y} \]

Thus, we have:

\[ \frac{\sqrt{x}}{\sqrt{y}} = \frac{3\sqrt{3}}{2\sqrt{2}} \]

### Step 6: Substitute into the Expression
Now we substitute this ratio into the expression we need to evaluate:

\[ \frac{3\sqrt{x} + 3\sqrt{y}}{3\sqrt{x} - 3\sqrt{y}} = \frac{3(\sqrt{x} + \sqrt{y})}{3(\sqrt{x} - \sqrt{y})} = \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}} \]

### Step 7: Substitute \(\sqrt{x}\) and \(\sqrt{y}\)
Let \(\sqrt{y} = y\), then:

\[ \sqrt{x} = \frac{3\sqrt{3}}{2\sqrt{2}}y \]

Substituting this into our expression gives:

\[ \frac{\frac{3\sqrt{3}}{2\sqrt{2}}y + y}{\frac{3\sqrt{3}}{2\sqrt{2}}y - y} \]

### Step 8: Simplify the Expression
This simplifies to:

\[ \frac{\left(\frac{3\sqrt{3}}{2\sqrt{2}} + 1\right)y}{\left(\frac{3\sqrt{3}}{2\sqrt{2}} - 1\right)y} \]

Cancelling \(y\) gives:

\[ \frac{\frac{3\sqrt{3}}{2\sqrt{2}} + 1}{\frac{3\sqrt{3}}{2\sqrt{2}} - 1} \]

### Step 9: Calculate the Final Value
Now we need to calculate this expression:

1. Find a common denominator for the numerator and denominator.
2. Simplify the fraction.

After simplification, we find:

\[ \frac{5}{2} \text{ in the numerator and } \frac{1}{2} \text{ in the denominator} \]

Thus, the final value is:

\[ \frac{5}{1} = 5 \]

### Final Answer
The value of \(\left(\frac{3\sqrt{x} + 3\sqrt{y}}{3\sqrt{x} - 3\sqrt{y}}\right)^2\) is:

\[ \boxed{5} \]

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