The equation `7y^(2) - 9x^(2) + 54x - 28y - 116 = 0 ` represents a _______

Answer

To determine the type of curve represented by the equation \( 7y^2 - 9x^2 + 54x - 28y - 116 = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 7y^2 - 9x^2 + 54x - 28y - 116 = 0 \]
Rearranging gives:
\[ 7y^2 - 28y - 9x^2 + 54x - 116 = 0 \]

### Step 2: Grouping Terms
Next, we group the \(y\) terms and the \(x\) terms:
\[ 7y^2 - 28y - 9x^2 + 54x = 116 \]

### Step 3: Completing the Square for \(y\)
To complete the square for the \(y\) terms:
\[ 7(y^2 - 4y) - 9x^2 + 54x = 116 \]
We take half of the coefficient of \(y\) (which is \(-4\)), square it, and add/subtract inside the parentheses:
\[ 7\left(y^2 - 4y + 4 - 4\right) - 9x^2 + 54x = 116 \]
This simplifies to:
\[ 7\left((y - 2)^2 - 4\right) - 9x^2 + 54x = 116 \]
Expanding gives:
\[ 7(y - 2)^2 - 28 - 9x^2 + 54x = 116 \]
Rearranging leads to:
\[ 7(y - 2)^2 - 9x^2 + 54x = 144 \]

### Step 4: Completing the Square for \(x\)
Now, we complete the square for the \(x\) terms:
\[ -9(x^2 - 6x) = -9\left((x - 3)^2 - 9\right) \]
This gives:
\[ -9(x - 3)^2 + 81 \]
Substituting back into the equation:
\[ 7(y - 2)^2 - 9(x - 3)^2 + 81 = 144 \]
Rearranging leads to:
\[ 7(y - 2)^2 - 9(x - 3)^2 = 63 \]

### Step 5: Dividing by 63
Dividing the entire equation by 63 gives:
\[ \frac{7(y - 2)^2}{63} - \frac{9(x - 3)^2}{63} = 1 \]
Simplifying further:
\[ \frac{(y - 2)^2}{9} - \frac{(x - 3)^2}{7} = 1 \]

### Step 6: Identifying the Conic Section
The equation is now in the standard form of a hyperbola:
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]
where \( (h, k) \) is the center of the hyperbola. Here, \( h = 3 \), \( k = 2 \), \( a^2 = 9 \), and \( b^2 = 7 \).

### Conclusion
Thus, the given equation represents a **hyperbola**.

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