The equation `y^(2)-x^(2)+2x-1=0`, represents

Options

A. a pair of st. lines
B. a circle
C. a parabola
D. an ellipse

Answer

To determine what the equation \( y^2 - x^2 + 2x - 1 = 0 \) represents, we can follow these steps:

### Step 1: Write the given equation
The given equation is:
\[ y^2 - x^2 + 2x - 1 = 0 \]

### Step 2: Rearrange the equation
We can rearrange the equation to fit the general form of a conic section:
\[ y^2 - x^2 + 2x - 1 = 0 \implies y^2 - (x^2 - 2x + 1) = 0 \]
This simplifies to:
\[ y^2 - (x - 1)^2 = 0 \]

### Step 3: Factor the equation
The equation can be factored as:
\[ (y - (x - 1))(y + (x - 1)) = 0 \]
This gives us two equations:
1. \( y - (x - 1) = 0 \) or \( y = x - 1 \)
2. \( y + (x - 1) = 0 \) or \( y = -x + 1 \)

### Step 4: Identify the type of conic section
The factored form indicates that the original equation represents two straight lines.

### Step 5: Conclusion
Thus, the equation \( y^2 - x^2 + 2x - 1 = 0 \) represents a pair of straight lines.

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