If `a + b -c = 12` and `a^(2) + b^(2) + c^(2) = 110`,
(p) ab + bc + ca = 34
(q) ab + bc - ca = 17
(r ) ab - bc + ca = 17
ab - bc - ca = 17
Which among the following relations is true ?

Options

A. r
B. q
C. p
D. s

Answer

To solve the problem step by step, we begin with the equations given:

1. **Equations Provided:**
- \( a + b - c = 12 \) (Equation 1)
- \( a^2 + b^2 + c^2 = 110 \) (Equation 2)

2. **Square Equation 1:**
We will square Equation 1:
\[ (a + b - c)^2 = 12^2 \]
Expanding the left side:
\[ a^2 + b^2 + c^2 + 2ab - 2ac - 2bc = 144 \]

3. **Substituting Equation 2:**
Now, substitute the value of \( a^2 + b^2 + c^2 \) from Equation 2 into the expanded equation:
\[ 110 + 2ab - 2ac - 2bc = 144 \]

4. **Rearranging the Equation:**
Rearranging gives us:
\[ 2ab - 2ac - 2bc = 144 - 110 \]
Simplifying the right side:
\[ 2ab - 2ac - 2bc = 34 \]

5. **Dividing by 2:**
Divide the entire equation by 2:
\[ ab - ac - bc = 17 \]

6. **Identifying the Correct Option:**
Now we compare this result with the options provided:
- (p) \( ab + bc + ca = 34 \)
- (q) \( ab + bc - ca = 17 \)
- (r) \( ab - bc + ca = 17 \)
- (s) \( ab - bc - ca = 17 \)

Based on our derived equation \( ab - ac - bc = 17 \), we can see that this matches option (s).

7. **Conclusion:**
Therefore, the correct relation among the options is:
\[ \text{Option (s) is true: } ab - bc - ca = 17 \]