Given that x : y :: y : z, we need to identify the correct statement among the following options:

a) xy = yz
b) y^2 = xz
c) xyz = x
d) zy = x

To find the correct statement, we can use the concept of ratios and proportions.

Understanding Ratios:
A ratio compares two or more quantities. It is expressed in the form of a fraction or with a colon (:). For example, if we say the ratio of boys to girls is 3:2, it means that for every 3 boys, there are 2 girls.

Using Double Colon (::):
When we have a double colon (::) in a ratio, it means that the two parts are in the same ratio as the other two parts. In other words, the two ratios are equivalent.

Applying the Given Ratio:
In the given ratio x : y :: y : z, we can rewrite it as x : y = y : z.

To find the correct statement, let's analyze each option:

a) xy = yz:
This statement is not correct because the product of x and y is not necessarily equal to the product of y and z. The ratio x : y is equivalent to the ratio y : z, but it doesn't mean that xy = yz. Therefore, option a) is incorrect.

b) y^2 = xz:
This statement is correct. Since x : y = y : z, we can rewrite it as x/y = y/z. Cross-multiplying, we get xy = y^2z. Rearranging the equation, we have y^2 = xz. Therefore, option b) is the correct statement.

c) xyz = x:
This statement is not correct. The given ratio x : y = y : z does not imply that xyz = x. Therefore, option c) is incorrect.

d) zy = x:
This statement is not correct either. The given ratio x : y = y : z does not imply that zy = x. Therefore, option d) is also incorrect.

Conclusion:
From the given options, the correct statement is b) y^2 = xz. This is the only statement that is consistent with the given ratio x : y :: y : z.