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 step by step, we start with the given equations:

1. **Given Equations:**
- \( a(a + b + c) = 45 \)
- \( b(a + b + c) = 75 \)
- \( c(a + b + c) = 105 \)

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

3. **Sum of \( a, b, c \):**
- Now substituting \( a, b, c \) back into \( S \):
\[ S = a + b + c = \frac{45}{S} + \frac{75}{S} + \frac{105}{S} \]
- Combine the fractions:
\[ S = \frac{45 + 75 + 105}{S} = \frac{225}{S} \]

4. **Multiply both sides by \( S \):**
- This gives:
\[ S^2 = 225 \]
- Taking the square root:
\[ S = 15 \quad (\text{since } S \text{ must be positive}) \]

5. **Finding values of \( a, b, c \):**
- Now substitute \( S = 15 \) back to find \( a, b, c \):
- \( a = \frac{45}{15} = 3 \)
- \( b = \frac{75}{15} = 5 \)
- \( c = \frac{105}{15} = 7 \)

6. **Calculate \( a^2 + b^2 + c^2 \):**
- Now we can find \( a^2 + b^2 + c^2 \):
\[ a^2 + b^2 + c^2 = 3^2 + 5^2 + 7^2 = 9 + 25 + 49 \]
- Adding these values:
\[ a^2 + b^2 + c^2 = 9 + 25 + 49 = 83 \]

7. **Final Answer:**
- Therefore, the value of \( a^2 + b^2 + c^2 \) is \( \boxed{83} \).