\[ \text { If } x+y+z=6 \sqrt{3} \text { and } x^2+y^2+z^2=36 . \text { Find } x: y: z \text {. } \]

Given:
– \(x + y + z = 6\sqrt{3}\)
– \(x^2 + y^2 + z^2 = 36\)

1. Use the identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\) to expand \((x + y + z)^2\):

\[ (x + y + z)^2 = (6\sqrt{3})^2 \]
\[ x^2 + y^2 + z^2 + 2(xy + yz + zx) = 108 \]
\[ 36 + 2(xy + yz + zx) = 108 \]
\[ 2(xy + yz + zx) = 72 \]
\[ xy + yz + zx = 36 \] (Equation A)

2. Comparing Equation A with \(x^2 + y^2 + z^2 = 36\):

\[ x^2 + y^2 + z^2 = xy + yz + zx \]

This implies that \(x = y = z\), as the sum of the squares of the variables is equal to the sum of their pairwise products.

3. Therefore, the ratio of \(x : y : z\) is \(1 : 1 : 1\).

Conclusion:
The ratio \(x : y : z\) is \(1 : 1 : 1\).