The mean of a,b and c is x. If `ab+bc+ca=0` what is the mean of `a^(2),b^(2)` and `c^(2)` ?

Options

A. `(x^(2))/(3)`
B. `x^(2)`
C. `3x^(2)`
D. `9x^(2)`

Answer

To solve the problem step by step, we need to find the mean of \( a^2, b^2, \) and \( c^2 \) given that the mean of \( a, b, \) and \( c \) is \( x \) and \( ab + bc + ca = 0 \).

### Step 1: Understand the Mean of \( a, b, c \)

The mean of \( a, b, c \) is given by:
\[ \text{Mean} = \frac{a + b + c}{3} = x \]
From this, we can express the sum of \( a, b, \) and \( c \):
\[ a + b + c = 3x \]

**Hint:** Remember that the mean is the sum of the values divided by the number of values.

### Step 2: Find the Mean of \( a^2, b^2, c^2 \)

The mean of \( a^2, b^2, c^2 \) is given by:
\[ \text{Mean} = \frac{a^2 + b^2 + c^2}{3} \]

**Hint:** We need to find \( a^2 + b^2 + c^2 \) to calculate this mean.

### Step 3: Use the Identity for Squaring a Sum

We know that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]
Substituting the values we have:
\[ (3x)^2 = a^2 + b^2 + c^2 + 2(0) \]
This simplifies to:
\[ 9x^2 = a^2 + b^2 + c^2 \]

**Hint:** This identity helps us relate the squares of the numbers to their sum and products.

### Step 4: Substitute into the Mean Formula

Now we can substitute \( a^2 + b^2 + c^2 \) into the mean formula:
\[ \text{Mean} = \frac{a^2 + b^2 + c^2}{3} = \frac{9x^2}{3} = 3x^2 \]

**Hint:** Make sure to simplify the fraction correctly.

### Conclusion

Thus, the mean of \( a^2, b^2, c^2 \) is:
\[ \text{Mean} = 3x^2 \]

**Final Answer:** The mean of \( a^2, b^2, \) and \( c^2 \) is \( 3x^2 \).