If `a ^(2) + b ^(2) = 169, ab = 60 ,` then `(a ^(2) - b ^(2))` is equal to :

Options

A. 149
B. 139
C. 119
D. 129

Answer

To solve the problem, we need to find the value of \( a^2 - b^2 \) given that \( a^2 + b^2 = 169 \) and \( ab = 60 \).

### Step-by-step Solution:

1. **Use the identity for \( a^2 + b^2 \)**:
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values:
\[ 169 = (a + b)^2 - 2 \times 60 \]
Simplifying this gives:
\[ 169 = (a + b)^2 - 120 \]
Therefore:
\[ (a + b)^2 = 169 + 120 = 289 \]

2. **Calculate \( a + b \)**:
Taking the square root of both sides:
\[ a + b = \sqrt{289} = 17 \]

3. **Use the identity for \( a^2 - b^2 \)**:
We know that:
\[ a^2 - b^2 = (a + b)(a - b) \]
We already have \( a + b = 17 \). Now we need to find \( a - b \).

4. **Use the identity for \( a^2 + b^2 \)** again to find \( a - b \)**:
We also know:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substituting the known values:
\[ 169 = (a - b)^2 + 120 \]
Simplifying this gives:
\[ (a - b)^2 = 169 - 120 = 49 \]

5. **Calculate \( a - b \)**:
Taking the square root of both sides:
\[ a - b = \sqrt{49} = 7 \]

6. **Calculate \( a^2 - b^2 \)**:
Now we can find \( a^2 - b^2 \):
\[ a^2 - b^2 = (a + b)(a - b) = 17 \times 7 = 119 \]

Thus, the value of \( a^2 - b^2 \) is **119**.