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

Answer

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

### Step 1: Identify coefficients
We start by comparing the given equation with the standard form of a conic section:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the equation \( 13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0 \), we can identify the coefficients:
- \( a = 13 \)
- \( b = 37 \)
- \( h = -9 \) (since \( 2h = -18 \))
- \( g = 1 \) (since \( 2g = 2 \))
- \( f = 7 \) (since \( 2f = 14 \))
- \( c = -2 \)

### Step 2: Calculate \( h^2 \) and \( ab \)
Next, we calculate \( h^2 \) and \( ab \):
\[ h^2 = (-9)^2 = 81 \]
\[ ab = 13 \times 37 = 481 \]

### Step 3: Check the conditions for conic sections
We need to check two conditions to determine the type of conic section:
1. \( h^2 < ab \)
2. The determinant \( \Delta \) is not equal to zero.

Since we have:
\[ h^2 = 81 < 481 = ab \]
This condition is satisfied.

### Step 4: Calculate the determinant \( \Delta \)
The determinant \( \Delta \) is given by:
\[ \Delta = abc + 2fgh - a f^2 - b g^2 - c h^2 \]
Substituting the values we found:
\[ \Delta = (13)(37)(-2) + 2(7)(1)(-9) - (13)(7^2) - (37)(1^2) - (-2)(-9^2) \]
Calculating each term:
- \( 13 \times 37 \times -2 = -962 \)
- \( 2 \times 7 \times 1 \times -9 = -126 \)
- \( 13 \times 49 = 637 \)
- \( 37 \times 1 = 37 \)
- \( -2 \times 81 = -162 \)

Now substituting back:
\[ \Delta = -962 - 126 - 637 - 37 - 162 \]
Calculating:
\[ \Delta = -962 - 126 - 637 - 37 - 162 = -1924 \]
Since \( \Delta \neq 0 \), this condition is also satisfied.

### Conclusion
Since both conditions are satisfied (\( h^2 < ab \) and \( \Delta \neq 0 \)), we conclude that the conic section represented by the equation \( 13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0 \) is an **ellipse**.