The equation ` 16x^(2)+y^(2) +8xy-74x-78y+212 =0` represents

Options

A. a circle
B. a parabola
C. an ellipse
D. hyperbola

Answer

To determine the type of conic section represented by the equation \( 16x^2 + y^2 + 8xy - 74x - 78y + 212 = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared to the general form of a conic section:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]
From the equation, we identify:
- \( a = 16 \)
- \( b = 1 \)
- \( 2h = 8 \) → \( h = 4 \)
- \( 2g = -74 \) → \( g = -37 \)
- \( 2f = -78 \) → \( f = -39 \)
- \( c = 212 \)

### Step 2: Calculate the determinant (Δ)
The determinant \( \Delta \) is calculated using the formula:
\[ \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \]
Substituting the values:
\[ \Delta = \begin{vmatrix} 16 & 4 & -37 \\ 4 & 1 & -39 \\ -37 & -39 & 212 \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant:
\[ \Delta = 16 \begin{vmatrix} 1 & -39 \\ -39 & 212 \end{vmatrix} - 4 \begin{vmatrix} 4 & -39 \\ -37 & 212 \end{vmatrix} - 37 \begin{vmatrix} 4 & 1 \\ -37 & -39 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} 1 & -39 \\ -39 & 212 \end{vmatrix} = (1)(212) - (-39)(-39) = 212 - 1521 = -1309 \)

2. \( \begin{vmatrix} 4 & -39 \\ -37 & 212 \end{vmatrix} = (4)(212) - (-39)(-37) = 848 - 1443 = -595 \)

3. \( \begin{vmatrix} 4 & 1 \\ -37 & -39 \end{vmatrix} = (4)(-39) - (1)(-37) = -156 + 37 = -119 \)

Now substituting back into the determinant:
\[ \Delta = 16(-1309) - 4(-595) - 37(-119) \]
Calculating:
\[ \Delta = -20944 + 2380 + 4403 = -14050 \]

### Step 4: Calculate \( h^2 - ab \)
Next, we calculate \( h^2 - ab \):
\[ h^2 - ab = 4^2 - (16)(1) = 16 - 16 = 0 \]

### Step 5: Determine the type of conic
Now we analyze the results:
- \( \Delta \neq 0 \) (since \(-14050 \neq 0\))
- \( h^2 - ab = 0 \)

According to the conic section classification:
- If \( \Delta \neq 0 \) and \( h^2 - ab = 0 \), the conic section is a **parabola**.

### Conclusion
The equation \( 16x^2 + y^2 + 8xy - 74x - 78y + 212 = 0 \) represents a **parabola**.

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