If `(a)/(b) = (25)/(6),` then the value of `(a ^(2) - b ^(2))/(a ^(2) + b ^(2))` is:

Options

A. A)`(589)/(661)`
B. B)`(589)/(651)`
C. C)`(625)/(36)`
D. D)`(661)/(589)`

Answer

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

\[ \frac{a}{b} = \frac{25}{6} \]

### Step 1: Express \(a\) in terms of \(b\)
From the ratio, we can express \(a\) as:

\[ a = \frac{25}{6} b \]

### Step 2: Substitute \(a\) in the expression
We need to find the value of:

\[ \frac{a^2 - b^2}{a^2 + b^2} \]

Substituting \(a = \frac{25}{6}b\) into the expression:

\[ a^2 = \left(\frac{25}{6}b\right)^2 = \frac{625}{36}b^2 \]

### Step 3: Calculate \(a^2 - b^2\) and \(a^2 + b^2\)
Now we can calculate \(a^2 - b^2\) and \(a^2 + b^2\):

1. **Calculate \(a^2 - b^2\)**:

\[ a^2 - b^2 = \frac{625}{36}b^2 - b^2 = \frac{625}{36}b^2 - \frac{36}{36}b^2 = \frac{625 - 36}{36}b^2 = \frac{589}{36}b^2 \]

2. **Calculate \(a^2 + b^2\)**:

\[ a^2 + b^2 = \frac{625}{36}b^2 + b^2 = \frac{625}{36}b^2 + \frac{36}{36}b^2 = \frac{625 + 36}{36}b^2 = \frac{661}{36}b^2 \]

### Step 4: Substitute back into the expression
Now substituting these results back into the expression:

\[ \frac{a^2 - b^2}{a^2 + b^2} = \frac{\frac{589}{36}b^2}{\frac{661}{36}b^2} \]

### Step 5: Simplify the expression
The \(b^2\) and \(36\) cancel out:

\[ \frac{589}{661} \]

### Final Answer
Thus, the value of \(\frac{a^2 - b^2}{a^2 + b^2}\) is:

\[ \frac{589}{661} \]

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