What conic does
`13x^2-18xy+37y^2+2x+14y-2=0` represent?

Answer

To determine what conic section the equation \( 13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0 \) represents, we will follow these steps:

### Step 1: Identify the coefficients
The general form of a conic section is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
From the given equation, we can identify the coefficients:
- \( A = 13 \)
- \( B = -18 \)
- \( C = 37 \)
- \( D = 2 \)
- \( E = 14 \)
- \( F = -2 \)

### Step 2: Calculate the discriminant
The discriminant \( \Delta \) for conic sections is calculated using the formula:
\[ \Delta = B^2 - 4AC \]
Substituting the values we have:
\[ \Delta = (-18)^2 - 4 \cdot 13 \cdot 37 \]
Calculating this gives:
\[ \Delta = 324 - 1964 = -1640 \]

### Step 3: Determine the type of conic
The type of conic can be determined based on the value of the discriminant:
- If \( \Delta < 0 \), the conic is an ellipse (or a circle if \( A = C \) and \( B = 0 \)).
- If \( \Delta = 0 \), the conic is a parabola.
- If \( \Delta > 0 \), the conic is a hyperbola.

Since \( \Delta = -1640 < 0 \), we conclude that the conic section represented by the given equation is an **ellipse**.

### Step 4: Conclusion
Thus, the conic represented by the equation \( 13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0 \) is an **ellipse**.

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