If x + 3y = - 3x -y , then `(x^(2))/( 2y^(2))` is equal to

Options

A. A)`(1)/(8)`
B. B)`(1)/(2)`
C. C)`(1)/(4)`
D. D)4

Answer

To solve the equation \( x + 3y = -3x - y \) and find the value of \( \frac{x^2}{2y^2} \), we can follow these steps:

### Step 1: Rearranging the equation
Start with the given equation:
\[ x + 3y = -3x - y \]
Now, let's move all terms involving \( x \) to one side and all terms involving \( y \) to the other side. We can add \( 3x \) and \( y \) to both sides:
\[ x + 3x + 3y + y = 0 \]
This simplifies to:
\[ 4x + 4y = 0 \]

### Step 2: Simplifying the equation
Now, we can factor out the common factor of 4:
\[ 4(x + y) = 0 \]
Dividing both sides by 4 gives:
\[ x + y = 0 \]
From this, we can express \( x \) in terms of \( y \):
\[ x = -y \]

### Step 3: Substituting into the expression
Now we need to find the value of \( \frac{x^2}{2y^2} \). Substituting \( x = -y \) into the expression gives:
\[ \frac{x^2}{2y^2} = \frac{(-y)^2}{2y^2} \]
This simplifies to:
\[ \frac{y^2}{2y^2} \]

### Step 4: Simplifying the fraction
Now, we can simplify the fraction:
\[ \frac{y^2}{2y^2} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{x^2}{2y^2} \) is:
\[ \frac{1}{2} \]

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