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Let's solve the problem step by step.

We are given:
1. \( x + y + z = 4\sqrt{6} \)
2. \( x^2 + y^2 + z^2 = 32 \)

We need to find the ratio \( x : y : z \).

### Step 1: Use the identity for squares
Recall the identity:
\[
(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
\]

Substituting the given values:
\[
(4\sqrt{6})^2 = 32 + 2(xy + yz + zx)
\]

\[
96 = 32 + 2(xy + yz + zx)
\]

\[
64 = 2(xy + yz + zx)
\]

\[
xy + yz + zx = 32
\]

### Step 2: Express in terms of \( x, y, z \)
We now have three equations:
1. \( x + y + z = 4\sqrt{6} \)
2. \( x^2 + y^2 + z^2 = 32 \)
3. \( xy + yz + zx = 32 \)

### Step 3: Test the given ratios
Let’s test each option to find which one satisfies all the equations.

#### Testing option (b): \( x : y : z = 1 : 2 : 3 \)
Assume \( x = k, y = 2k, z = 3k \). 

1. Substituting into \( x + y + z \):
   \[
   k + 2k + 3k = 6k
   \]
   We have \( 6k = 4\sqrt{6} \).
   \[
   k = \frac{4\sqrt{6}}{6} = \frac{2\sqrt{6}}{3}
   \]

2. Substitute \( k = \frac{2\sqrt{6}}{3} \) into \( x^2 + y^2 + z^2 \):
   \[
   x^2 = \left(\frac{2\sqrt{6}}{3}\right)^2 = \frac{24}{9} = \frac{8}{3}
   \]
   \[
   y^2 = \left(2 \cdot \frac{2\sqrt{6}}{3}\right)^2 = \frac{96}{9} = \frac{32}{3}
   \]
   \[
   z^2 = \left(3 \cdot \frac{2\sqrt{6}}{3}\right)^2 = 24
   \]

   Adding these:
   \[
   x^2 + y^2 + z^2 = \frac{8}{3} + \frac{32}{3} + 24 = \frac{40}{3} + 24 = 32
   \]

Since all conditions are satisfied, the correct ratio is:
\[
\boxed{1 : 2 : 3}
\]

Thus, the answer is (b).

---

Would you like further details or have any questions? Here are some related questions:

1. How would the solution change if \( x^2 + y^2 + z^2 \) had a different value?
2. Can you solve a similar problem with different values for \( x + y + z \) and \( x^2 + y^2 + z^2 \)?
3. What if \( x, y, z \) are not integers—how do we solve for their values?
4. How can we derive the formulas for sums of products of variables in three-variable equations?
5. What approach would you take if given \( x^3 + y^3 + z^3 \) instead?

**Tip:** Always check whether given equations can be transformed using algebraic identities like expansion of squares to simplify the problem.

If x + y + z = 4√6 and x² + y² + z² = 32 then find the ratio x:y:z. Options: (a) 1:1:1, (b) 1:2:3, (c) 1:3:4, (d) 3:2:1

Solve the problem where x + y + z = 4√6 and x² + y² + z² = 32 to find the ratio of x, y, and z. This involves algebraic identities and solving a system of equations. Suitable for students in Grades 9-11.

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