If the average of a, b, c is 'm' and (ab + bc + ca) = 0, then the average value of $a^2 + b^2 + c^2$ is :

Q: If the average of a, b, c is 'm' and (ab + bc + ca) = 0, then the average value of $a^2 + b^2 + c^2$ is :

To determine the average value of $a^2 + b^2 + c^2$ given that the average of $a, b, c$ is $m$ and $(ab + bc + ca) = 0$, we can follow these steps: 1. **Understanding the Given Conditions:** - The average of $a, b, c$ is $m$, which implies: \[ \frac{a + b + c}{3} = m \quad \Rightarrow \quad a + b + c = 3m \] - The condition $(ab + bc + ca) = 0$ is also given. 2. **Expression for $a^2 + b^2 + c^2$:** - We know the identity: \[ a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) \] - Substituting the given conditions: \[ a^2 + b^2 + c^2 = (3m)^2 - 2 \times 0 = 9m^2 \] 3. **Average Value of $a^2 + b^2 + c^2$:** - Since we are looking for the average value of $a^2 + b^2 + c^2$, we divide by 3: \[ \text{Average of } a^2 + b^2 + c^2 = \frac{9m^2}{3} = 3m^2 \] 4. **Conclusion:** - Therefore, the average value of $a^2 + b^2 + c^2$ is $3m^2$, which corresponds to Option 2. 5. **Explanation of Incorrect Options:** - **Option 1 ($m^2$):** This option suggests that the average of $a^2 + b^2 + c^2$ is simply $m^2$, which does not account for the squared sum of $a, b, c$ and the given condition $(ab + bc + ca) = 0$. - **Option 3 ($6m^2$):** This option overestimates the average by assuming a different relationship or miscalculation in the identity used. - **Option 4 ($9m^2$):** This option represents the total sum $a^2 + b^2 + c^2$ without dividing by 3 to find the average. Thus, the correct answer is indeed Option 2, $3m^2$, as it accurately reflects the average value of $a^2 + b^2 + c^2$ given the conditions.

Question

Accepted Answer

To determine the average value of $a^2 + b^2 + c^2$ given that the average of $a, b, c$ is $m$ and $(ab + bc + ca) = 0$, we can follow these steps:

1. **Understanding the Given Conditions:**
   - The average of $a, b, c$ is $m$, which implies:
     $$
     \frac{a + b + c}{3} = m \quad \Rightarrow \quad a + b + c = 3m
     $$
   - The condition $(ab + bc + ca) = 0$ is also given.

2. **Expression for $a^2 + b^2 + c^2$:**
   - We know the identity:
     $$
     a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
     $$
   - Substituting the given conditions:
     $$
     a^2 + b^2 + c^2 = (3m)^2 - 2 \times 0 = 9m^2
     $$

3. **Average Value of $a^2 + b^2 + c^2$:**
   - Since we are looking for the average value of $a^2 + b^2 + c^2$, we divide by 3:
     $$
     \text{Average of } a^2 + b^2 + c^2 = \frac{9m^2}{3} = 3m^2
     $$

4. **Conclusion:**
   - Therefore, the average value of $a^2 + b^2 + c^2$ is $3m^2$, which corresponds to Option 2.

5. **Explanation of Incorrect Options:**
   - **Option 1 ($m^2$):** This option suggests that the average of $a^2 + b^2 + c^2$ is simply $m^2$, which does not account for the squared sum of $a, b, c$ and the given condition $(ab + bc + ca) = 0$.
   - **Option 3 ($6m^2$):** This option overestimates the average by assuming a different relationship or miscalculation in the identity used.
   - **Option 4 ($9m^2$):** This option represents the total sum $a^2 + b^2 + c^2$ without dividing by 3 to find the average.

Thus, the correct answer is indeed Option 2, $3m^2$, as it accurately reflects the average value of $a^2 + b^2 + c^2$ given the conditions.