If `(a+b):(a-b)=3:2` the `(a^(2)-b^(2)):(a^(2)+b^(2))` equals:

Options

A. `5:13`
B. `12:13`
C. `9:14`
D. None of these

Answer

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

\[ (a+b):(a-b) = 3:2 \]

### Step 1: Set up the equation from the ratio
From the ratio, we can express it as:

\[ \frac{a+b}{a-b} = \frac{3}{2} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 2(a+b) = 3(a-b) \]

### Step 3: Expand both sides
Expanding both sides results in:

\[ 2a + 2b = 3a - 3b \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to group like terms:

\[ 2a + 2b + 3b = 3a \]

This simplifies to:

\[ 2a + 5b = 3a \]

### Step 5: Isolate \(a\)
Now, we isolate \(a\):

\[ 5b = 3a - 2a \]

This simplifies to:

\[ 5b = a \]

### Step 6: Express \(a\) in terms of \(b\)
Thus, we have:

\[ a = 5b \]

### Step 7: Substitute \(a\) into the expression we need to find
We need to find the ratio:

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

Substituting \(a = 5b\):

\[ \frac{(5b)^2 - b^2}{(5b)^2 + b^2} \]

### Step 8: Simplify the expression
Calculating \(a^2\) and \(b^2\):

\[ \frac{25b^2 - b^2}{25b^2 + b^2} = \frac{24b^2}{26b^2} \]

### Step 9: Simplify the ratio
Now, we can simplify the ratio:

\[ \frac{24}{26} = \frac{12}{13} \]

### Final Answer
Thus, the final answer is:

\[ \frac{a^2 - b^2}{a^2 + b^2} = \frac{12}{13} \]

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