If `a(a + b +c) =45, b(a +b+c)=75 and c (a +b+c) = 105`, then what is the value of `(a^(2) + b^(2) + c^(2))`?

Options

A. 75
B. 83
C. 217
D. 225

Answer

To solve the problem, we will follow these steps:

1. **Given Equations**: We have three equations:
\[ a(a + b + c) = 45 \quad (1) \]
\[ b(a + b + c) = 75 \quad (2) \]
\[ c(a + b + c) = 105 \quad (3) \]

2. **Let \( S = a + b + c \)**: From the equations, we can express \( S \) in terms of \( a \), \( b \), and \( c \):
- From (1): \( aS = 45 \) → \( S = \frac{45}{a} \)
- From (2): \( bS = 75 \) → \( S = \frac{75}{b} \)
- From (3): \( cS = 105 \) → \( S = \frac{105}{c} \)

3. **Equating the expressions for \( S \)**:
\[ \frac{45}{a} = \frac{75}{b} = \frac{105}{c} \]

4. **Let \( k = S \)**: We can express \( a \), \( b \), and \( c \) in terms of \( k \):
\[ a = \frac{45}{k}, \quad b = \frac{75}{k}, \quad c = \frac{105}{k} \]

5. **Finding \( S \)**: Now, substitute \( a \), \( b \), and \( c \) back into the equation for \( S \):
\[ S = a + b + c = \frac{45}{k} + \frac{75}{k} + \frac{105}{k} = \frac{45 + 75 + 105}{k} = \frac{225}{k} \]

6. **Setting the two expressions for \( S \) equal**:
\[ k = \frac{225}{k} \implies k^2 = 225 \implies k = 15 \quad (\text{since } k > 0) \]

7. **Finding \( a \), \( b \), and \( c \)**:
\[ a = \frac{45}{15} = 3, \quad b = \frac{75}{15} = 5, \quad c = \frac{105}{15} = 7 \]

8. **Finding \( a^2 + b^2 + c^2 \)**:
\[ a^2 + b^2 + c^2 = 3^2 + 5^2 + 7^2 = 9 + 25 + 49 = 83 \]

Thus, the value of \( a^2 + b^2 + c^2 \) is \( \boxed{83} \).