If $x^2 + 4y^2 = 4xy$, then $x : y$ is

### Concept & Algebraic Identities The problem relies on recognizing perfect square trinomials. By rearranging the terms, we can form a perfect square using the identity: $$ (a - b)^2 = a^2 - 2ab + b^2 $$ ### Step-by-Step Solution 1. Start with the given equation: $x^2 + 4y^2 = 4xy$ 2. Move all terms to one side of the equation to set it to zero: $x^2 - 4xy + 4y^2 = 0$ 3. Recognize that this is a perfect square. It can be factored as: $(x - 2y)^2 = 0$ 4. Take the square root of both sides: $x - 2y = 0$ 5. Isolate $x$: $x = 2y$ 6. Find the ratio $x : y$ by dividing both sides by $y$: $\frac{x}{y} = \frac{2}{1}$ ### Exam Strategy & Shortcut When you see a quadratic equation with two variables set equal to each other in this format, immediately move all terms to one side. If it forms a perfect square $(ax - by)^2 = 0$, you can bypass the factoring steps and directly conclude that $ax = by$. ### Common Pitfall A common mistake is trying to divide the entire equation by $xy$ right away, which complicates the algebra by creating rational expressions instead of a clean, recognizable perfect square. ### Final Answer Therefore, the correct answer is **$2 : 1$**.