If `a,b,c` are sides an acute angle triangle satisfying `a^(2)+b^(2)+c^(2)=6` then `(ab+bc+ca)` can be equal to

Options

A. 3
B. 4
C. 5
D. 7

Explanation

To solve the problem, we need to find the possible values of \( ab + bc + ca \) given that \( a, b, c \) are the sides of an acute triangle and satisfy the equation \( a^2 + b^2 + c^2 = 6 \).

### Step-by-Step Solution:

1. **Understanding the properties of an acute triangle**:
In an acute triangle, the square of each side must be less than the sum of the squares of the other two sides. This gives us three inequalities:
\[ a^2 < b^2 + c^2, \quad b^2 < a^2 + c^2, \quad c^2 < a^2 + b^2 \]

2. **Using the given equation**:
We know that:
\[ a^2 + b^2 + c^2 = 6 \]
From this, we can derive:
\[ b^2 + c^2 = 6 - a^2, \quad a^2 + c^2 = 6 - b^2, \quad a^2 + b^2 = 6 - c^2 \]

3. **Substituting into the inequalities**:
Using the inequalities from step 1, we can substitute:
\[ a^2 < 6 - a^2 \implies 2a^2 < 6 \implies a^2 < 3 \]
Similarly, we can derive:
\[ b^2 < 3 \quad \text{and} \quad c^2 < 3 \]

4. **Finding bounds for \( ab + bc + ca \)**:
We know from the Cauchy-Schwarz inequality that:
\[ (ab + bc + ca) \leq \frac{(a+b+c)^2}{3} \]
To find \( a + b + c \), we can use the fact that:
\[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting \( a^2 + b^2 + c^2 = 6 \):
\[ (a+b+c)^2 = 6 + 2(ab + ac + bc) \]

5. **Setting up the inequalities**:
From the previous steps, we can derive:
\[ ab + ac + bc = \frac{(a+b+c)^2 - 6}{2} \]
We need to find the maximum and minimum values of \( ab + ac + bc \).

6. **Using AM-GM inequality**:
By the AM-GM inequality:
\[ \frac{a^2 + b^2 + c^2}{3} \geq \sqrt[3]{a^2b^2c^2} \]
Since \( a^2 + b^2 + c^2 = 6 \):
\[ 2 \geq \sqrt[3]{a^2b^2c^2} \implies 8 \geq a^2b^2c^2 \implies 2 \geq abc \]

7. **Finding the range of \( ab + ac + bc \)**:
From the inequalities derived:
\[ ab + ac + bc < 6 \]
And from the earlier steps, we also found that:
\[ ab + ac + bc > 3 \]

8. **Conclusion**:
Therefore, the value of \( ab + ac + bc \) can lie between 3 and 6. The possible integer values are 4 and 5.

### Final Answer:
Thus, \( ab + ac + bc \) can be equal to **4 or 5**.