The equation `x^(2)y^(2)-9y^(2)-6x^(2)y+54y=0` represents

Options

A. a pair of straight lines and a circle
B. a pair of straight lines and a parabola
C. a set of four straight lines forming a square
D. None of these

Explanation

To solve the equation \( x^2y^2 - 9y^2 - 6x^2y + 54y = 0 \) and determine what it represents, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x^2y^2 - 9y^2 - 6x^2y + 54y = 0 \]

### Step 2: Factoring the Equation
We can factor the equation by grouping terms. Notice that we can group the first two terms and the last two terms:
\[ y^2(x^2 - 9) - 6y(x^2 - 9) = 0 \]
Now, we can factor out \( (x^2 - 9) \):
\[ (x^2 - 9)(y^2 - 6y) = 0 \]

### Step 3: Further Factoring
Next, we can factor \( x^2 - 9 \) and \( y^2 - 6y \):
\[ (x + 3)(x - 3)(y(y - 6)) = 0 \]

### Step 4: Finding the Roots
Setting each factor to zero gives us:
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x - 3 = 0 \) → \( x = 3 \)
3. \( y = 0 \)
4. \( y - 6 = 0 \) → \( y = 6 \)

### Step 5: Identifying the Lines
From the roots, we can see that we have:
- Two vertical lines: \( x = -3 \) and \( x = 3 \)
- Two horizontal lines: \( y = 0 \) and \( y = 6 \)

### Conclusion
The equation represents a set of four straight lines:
- \( x = -3 \)
- \( x = 3 \)
- \( y = 0 \)
- \( y = 6 \)

These lines can be visualized as forming a rectangle (or square) on the coordinate plane.

### Final Answer
The equation represents a set of four straight lines forming a square.

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