If the square of the sum of two numbers is equal to 4 times of their product, then the ratio of these numbers is:

Options

A. `2:1`
B. `1:3`
C. `1:1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of two numbers \( x \) and \( y \) given that the square of their sum is equal to four times their product. Let's break this down step by step.

### Step-by-Step Solution:

1. **Set Up the Equation**:
We start with the given condition:
\[ (x + y)^2 = 4xy \]

2. **Expand the Left Side**:
We expand the left side of the equation:
\[ x^2 + 2xy + y^2 = 4xy \]

3. **Rearrange the Equation**:
Next, we rearrange the equation by moving all terms to one side:
\[ x^2 + y^2 + 2xy - 4xy = 0 \]
This simplifies to:
\[ x^2 + y^2 - 2xy = 0 \]

4. **Recognize a Perfect Square**:
Notice that the left side can be factored as a perfect square:
\[ (x - y)^2 = 0 \]

5. **Take the Square Root**:
Taking the square root of both sides gives us:
\[ x - y = 0 \]

6. **Solve for the Ratio**:
From \( x - y = 0 \), we find that:
\[ x = y \]
Therefore, the ratio of \( x \) to \( y \) is:
\[ \frac{x}{y} = 1 \]
This means the ratio of the two numbers is \( 1:1 \).

### Final Answer:
The ratio of the two numbers is \( 1:1 \).