For what value of ` 'y' , x^(2) + (1)/(12) x + y^(2)` is a perfect square ?

Options

A. `(1)/(24)`
B. `(1)/(12)`
C. `(1)/(6)`
D. `(1)/(3)`

Answer

To determine the value of \( y \) for which the expression \( x^2 + \frac{1}{12}x + y^2 \) is a perfect square, we can follow these steps:

### Step 1: Identify the form of a perfect square
A perfect square trinomial can be expressed in the form \( (a + b)^2 = a^2 + 2ab + b^2 \). In our case, we can compare our expression to this form.

### Step 2: Rewrite the expression
We have:
\[ x^2 + \frac{1}{12}x + y^2 \]
We can identify \( a = x \) and \( b = y \). Thus, we want to express \( \frac{1}{12}x \) in terms of \( 2ab \).

### Step 3: Set up the equation
From the perfect square form, we know:
\[ 2ab = \frac{1}{12}x \]
Substituting \( a = x \) and \( b = y \):
\[ 2xy = \frac{1}{12}x \]

### Step 4: Solve for \( y \)
To isolate \( y \), we can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 2y = \frac{1}{12} \]
Now, divide both sides by 2:
\[ y = \frac{1}{24} \]

### Step 5: Conclusion
Thus, the value of \( y \) for which \( x^2 + \frac{1}{12}x + y^2 \) is a perfect square is:
\[ \boxed{\frac{1}{24}} \]