If the expression ` (x^(2))/(y^(2)) + tx + (y^(2))/( 4) ` is a perfect square then the values of t is

Options

A. `pm1`
B. `pm2`
C. 0
D. `pm 3`

Answer

To determine the value of \( t \) such that the expression

\[ \frac{x^2}{y^2} + tx + \frac{y^2}{4} \]

is a perfect square, we will follow these steps:

### Step 1: Identify the structure of a perfect square
A perfect square trinomial can be expressed in the form \( (a + b)^2 \), which expands to \( a^2 + 2ab + b^2 \).

### Step 2: Assign values to \( a \) and \( b \)
In our expression, we can identify:
- \( a = \frac{x}{y} \)
- \( b = \frac{y}{2} \)

### Step 3: Expand the perfect square
Using the identified values, we can expand \( (a + b)^2 \):
\[ (a + b)^2 = \left(\frac{x}{y} + \frac{y}{2}\right)^2 = \left(\frac{x}{y}\right)^2 + 2\left(\frac{x}{y}\right)\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)^2 \]

### Step 4: Calculate each term
Calculating each term:
1. \( \left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2} \)
2. \( 2\left(\frac{x}{y}\right)\left(\frac{y}{2}\right) = \frac{2xy}{2y} = x \)
3. \( \left(\frac{y}{2}\right)^2 = \frac{y^2}{4} \)

### Step 5: Combine the terms
Combining these, we get:
\[ \frac{x^2}{y^2} + x + \frac{y^2}{4} \]

### Step 6: Compare with the original expression
Now, we compare this with the original expression:
\[ \frac{x^2}{y^2} + tx + \frac{y^2}{4} \]
To make these two expressions equal, we can see that \( t \) must equal \( 1 \) because the coefficient of \( x \) in the expanded perfect square is \( 1 \).

### Step 7: Conclusion
Thus, the value of \( t \) that makes the expression a perfect square is:
\[ t = 1 \]