Understanding the Problem: Mean and Sum of Squares

The question asks us to find the mean of the squares of three numbers, p, q, and r, given their mean and a specific condition involving their products.

We are given two pieces of information:

1. The mean of p, q, and r is M.
2. The condition pq + qr + rp = 0 is satisfied.

We need to calculate the mean of p², q², and r².

Calculating the Sum of the Numbers

The mean of p, q, and r is given by the formula:

\( \text{Mean} = \frac{\text{Sum of numbers}}{\text{Number of terms}} \)

In this case, the mean is M and the numbers are p, q, and r (3 terms). So,

\( M = \frac{p+q+r}{3} \)

From this equation, we can find the sum of the numbers:

\( p+q+r = 3M \)

Relating Sum of Numbers to Sum of Squares

We need to find the mean of p², q², and r². This requires us to find the sum p² + q² + r². We can use an algebraic identity that relates the sum of numbers to the sum of their squares and the sum of their products taken two at a time.

The identity is:

\( (p+q+r)^2 = p^2 + q^2 + r^2 + 2(pq + qr + rp) \)

We know the values for \( (p+q+r) \) and \( (pq + qr + rp) \) from the given information.

• Substitute \( (p+q+r) = 3M \)
• Substitute \( (pq + qr + rp) = 0 \)

Plugging these values into the identity:

\( (3M)^2 = p^2 + q^2 + r^2 + 2(0) \)

\( 9M^2 = p^2 + q^2 + r^2 + 0 \)

\( 9M^2 = p^2 + q^2 + r^2 \)

So, the sum of the squares of the numbers is \( 9M^2 \).

Calculating the Mean of the Squares

Now that we have the sum of the squares, we can find their mean. The mean of p², q², and r² is:

\( \text{Mean of squares} = \frac{p^2 + q^2 + r^2}{3} \)

Substitute the value of \( (p^2 + q^2 + r^2) \) that we just found:

\( \text{Mean of squares} = \frac{9M^2}{3} \)

\( \text{Mean of squares} = 3M^2 \)

Therefore, the mean of p², q², r² is \( 3M^2 \).

Summary of Steps

1. Used the given mean of p, q, r to find the sum (p+q+r) in terms of M.
2. Used the algebraic identity \( (p+q+r)^2 = p^2 + q^2 + r^2 + 2(pq + qr + rp) \).
3. Substituted the values of \( (p+q+r) \) and \( (pq + qr + rp) \) into the identity.
4. Solved for \( (p^2 + q^2 + r^2) \).
5. Calculated the mean of p², q², r² by dividing the sum of squares by 3.

Revision Table: Key Concepts

Additional Information: The Significance of pq + qr + rp = 0

The condition pq + qr + rp = 0 is quite specific. It means that the sum of pairwise products of the numbers is zero. Geometrically, if p, q, and r were roots of a cubic polynomial \( x^3 - (p+q+r)x^2 + (pq+qr+rp)x - pqr = 0 \), this condition simplifies the polynomial to \( x^3 - (p+q+r)x^2 - pqr = 0 \). In the context of algebraic identities, this condition directly simplifies the expansion of \( (p+q+r)^2 \), as shown in the solution, making the calculation of the sum of squares straightforward.

This problem demonstrates how understanding basic algebraic identities and the definition of mean can be used together to solve problems involving relationships between sums and sums of squares.