If  $$8x^2 + 2y^2 + 18z^2 - 4xy - 6yz - 12zx = 0$$​, then find x : y : z.

Correct option is D

Given
Equation: $$8x^2 + 2y^2 + 18z^2 - 4xy - 6yz - 12zx = 0$$​​
Formula Used
​$$(a - b)^2 = a^2 - 2ab + b^2$$​​
Solution
Rewrite the given equation as a sum of perfect squares:
​$$(4x^2 - 4xy + y^2) + (y^2 - 6yz + 9z^2) + (9z^2 - 12zx + 4x^2) = 0\\(2x - y)^2 + (y - 3z)^2 + (3z - 2x)^2 = 0$$​​
Since the sum of squares is zero, each term must be zero:
​$$2x - y = 0 \Rightarrow 2x = y \\y - 3z = 0 \Rightarrow y = 3z\\3z - 2x = 0 \Rightarrow 3z = 2x\\$$​​
Therefore, 2x = y = 3z = k
​$$x = \frac{k}{2}, y = k, z = \frac{k}{3}$$​​
Ratio x : y : z = $$\frac{1}{2} : 1 : \frac{1}{3}$$​​
Multiply by 6 to remove denominators:
x : y : z = 3 : 6 : 2
Final Answer
So the correct answer is (d)