The conic represented by the equation `x^(2)+y^(2)-2xy+20x+10=0,` is

Options

A. Pair of straight lines
B. Circle
C. Paraabola
D. Ellipse

Explanation

To determine the type of conic represented by the equation \( x^2 + y^2 - 2xy + 20x + 10 = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The general form of a conic is given by:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 1 \) (coefficient of \( y^2 \))
- \( 2h = -2 \) (coefficient of \( xy \)), thus \( h = -1 \)
- \( 2g = 20 \) (coefficient of \( x \)), thus \( g = 10 \)
- \( 2f = 0 \) (coefficient of \( y \)), thus \( f = 0 \)
- \( c = 10 \)

### Step 2: Calculate the discriminant \( \Delta \)
The discriminant \( \Delta \) is given by the formula:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - c \]
Substituting the values we found:
\[ \Delta = (1)(1)(10) + 2(0)(-1)(10) - (1)(0^2) - (1)(10^2) - 10 \]
Calculating each term:
- \( abc = 10 \)
- \( 2fgh = 0 \)
- \( af^2 = 0 \)
- \( bg^2 = 100 \)
- \( c = 10 \)

Now, substituting these values into the discriminant:
\[ \Delta = 10 + 0 - 0 - 100 - 10 = 10 - 100 - 10 = -100 \]

### Step 3: Calculate \( h^2 - ab \)
Next, we calculate \( h^2 - ab \):
\[ h^2 - ab = (-1)^2 - (1)(1) = 1 - 1 = 0 \]

### Step 4: Determine the type of conic
We have:
- \( \Delta \neq 0 \) (specifically, \( \Delta = -100 \))
- \( h^2 - ab = 0 \)

According to the conditions for conics:
- If \( \Delta \neq 0 \) and \( h^2 - ab = 0 \), the conic represents a **parabola**.

### Conclusion
Thus, the conic represented by the equation \( x^2 + y^2 - 2xy + 20x + 10 = 0 \) is a **parabola**.

---