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
We start by rewriting the given equation in the standard form of a conic section:

\[ x^2 + y^2 - 2xy + 20x + 10 = 0 \]

This can be compared with the general second-degree equation:

\[ Ax^2 + By^2 + 2Hxy + 2Gx + 2Fy + C = 0 \]

From our equation, we can identify:
- \( A = 1 \)
- \( B = 1 \)
- \( H = -1 \)
- \( G = 10 \)
- \( F = 0 \) (since there is no \( y \) term)
- \( C = 10 \)

### Step 2: Calculate the discriminant
To determine the type of conic, we calculate the discriminant \( \Delta \):

\[ \Delta = ABC + 2FGH - AF^2 - BG^2 - CH^2 \]

Substituting the values we identified:
- \( A = 1 \)
- \( B = 1 \)
- \( C = 10 \)
- \( F = 0 \)
- \( G = 10 \)
- \( H = -1 \)

Now substituting into the discriminant formula:

\[ \Delta = (1)(1)(10) + 2(0)(10)(-1) - (1)(0)^2 - (1)(10)^2 - (10)(-1)^2 \]

Calculating each term:

\[ \Delta = 10 + 0 - 0 - 100 - 10 \]
\[ \Delta = 10 - 100 - 10 = -100 \]

### Step 3: Determine the type of conic
The sign of the discriminant \( \Delta \) helps us classify the conic:
- If \( \Delta > 0 \), it is a hyperbola.
- If \( \Delta = 0 \), it is a parabola.
- If \( \Delta < 0 \), it is an ellipse or a circle.

Since \( \Delta = -100 < 0 \), the conic represented by the equation is an **ellipse**.

### Summary
The conic represented by the equation \( x^2 + y^2 - 2xy + 20x + 10 = 0 \) is an **ellipse**.

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