If ` x^(2) + 9 y^(2) = 6 xy ` then x : y is

Options

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

Answer

To solve the equation \( x^2 + 9y^2 = 6xy \) and find the ratio \( x : y \), we can follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation:
\[ x^2 + 9y^2 - 6xy = 0 \]

### Step 2: Recognize the quadratic form
This equation can be recognized as a quadratic in terms of \( x \):
\[ x^2 - 6xy + 9y^2 = 0 \]

### Step 3: Factor the quadratic
Notice that the quadratic can be factored using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ (x - 3y)^2 = 0 \]

### Step 4: Solve for \( x \)
Setting the factored form equal to zero gives:
\[ x - 3y = 0 \]
From this, we can solve for \( x \):
\[ x = 3y \]

### Step 5: Find the ratio \( x : y \)
To express the ratio \( x : y \), we substitute \( x \) with \( 3y \):
\[ x : y = 3y : y \]
This simplifies to:
\[ x : y = 3 : 1 \]

### Final Answer
Thus, the ratio \( x : y \) is \( 3 : 1 \).

---