If `(a +b):(a-b)= 5 : 3` then find `(a^2 + b^2):(a^2 -b^2)`.

Options

A. `17 : 15`
B. `25 : 9`
C. `4 : 1`
D. `16 : 1`

Answer

To solve the problem, we need to find the ratio \((a^2 + b^2):(a^2 - b^2)\) given that \((a + b):(a - b) = 5:3\).

### Step-by-Step Solution:

1. **Set up the initial ratio**:
We know that \((a + b):(a - b) = 5:3\). This can be expressed as:
\[ \frac{a + b}{a - b} = \frac{5}{3} \]

2. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives us:
\[ 3(a + b) = 5(a - b) \]

3. **Expand both sides**:
Expanding both sides results in:
\[ 3a + 3b = 5a - 5b \]

4. **Rearrange the equation**:
To isolate terms involving \(a\) and \(b\), we can rearrange the equation:
\[ 3a + 3b + 5b = 5a \]
Simplifying gives:
\[ 3a + 8b = 5a \]

5. **Isolate \(a\)**:
Rearranging further gives:
\[ 8b = 5a - 3a \]
Thus:
\[ 8b = 2a \quad \Rightarrow \quad a = 4b \]

6. **Substitute \(a\) in terms of \(b\)**:
Now we can express \(a\) in terms of \(b\):
\[ a = 4b \]

7. **Find \(a^2 + b^2\) and \(a^2 - b^2\)**:
Now we calculate \(a^2 + b^2\) and \(a^2 - b^2\):
\[ a^2 + b^2 = (4b)^2 + b^2 = 16b^2 + b^2 = 17b^2 \]
\[ a^2 - b^2 = (4b)^2 - b^2 = 16b^2 - b^2 = 15b^2 \]

8. **Form the ratio**:
Now we can form the ratio \((a^2 + b^2):(a^2 - b^2)\):
\[ (a^2 + b^2):(a^2 - b^2) = 17b^2 : 15b^2 \]

9. **Simplify the ratio**:
The \(b^2\) cancels out:
\[ 17 : 15 \]

### Final Answer:
Thus, the required ratio \((a^2 + b^2):(a^2 - b^2)\) is:
\[ \boxed{17 : 15} \]