The axis of the parabola `9y^2-16x-12y-57 = 0` is

Options

A. `3y=2`
B. `x+3y=3`
C. `2x=3`
D. `y=3`

Explanation

To find the axis of the parabola given by the equation \(9y^2 - 16x - 12y - 57 = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to isolate the terms involving \(y\):
\[ 9y^2 - 12y - 16x - 57 = 0 \]

### Step 2: Dividing by 9
Next, divide the entire equation by 9 to simplify it:
\[ y^2 - \frac{12}{9}y - \frac{16}{9}x - \frac{57}{9} = 0 \]
This simplifies to:
\[ y^2 - \frac{4}{3}y - \frac{16}{9}x - \frac{19}{3} = 0 \]

### Step 3: Grouping \(y\) Terms
Now, move the \(x\) terms to the right side:
\[ y^2 - \frac{4}{3}y = \frac{16}{9}x + \frac{19}{3} \]

### Step 4: Completing the Square
To complete the square for the \(y\) terms, we take the coefficient of \(y\) (which is \(-\frac{4}{3}\)), halve it, and square it:
\[ \left(-\frac{4}{6}\right)^2 = \frac{4}{9} \]
Add \(\frac{4}{9}\) to both sides:
\[ y^2 - \frac{4}{3}y + \frac{4}{9} = \frac{16}{9}x + \frac{19}{3} + \frac{4}{9} \]

### Step 5: Simplifying the Right Side
Now simplify the right side:
\[ y - \frac{2}{3} = \frac{16}{9}x + \left(\frac{19}{3} + \frac{4}{9}\right) \]
To combine the fractions on the right, convert \(\frac{19}{3}\) to have a denominator of 9:
\[ \frac{19}{3} = \frac{57}{9} \]
Thus, we have:
\[ y - \frac{2}{3} = \frac{16}{9}x + \frac{57 + 4}{9} = \frac{16}{9}x + \frac{61}{9} \]

### Step 6: Final Form of the Parabola
Rearranging gives us:
\[ \left(y - \frac{2}{3}\right)^2 = \frac{16}{9}x + \frac{61}{9} \]
Factoring out \(\frac{16}{9}\) from the right side:
\[ \left(y - \frac{2}{3}\right)^2 = \frac{16}{9}\left(x + \frac{61}{16}\right) \]

### Step 7: Identifying the Axis
This is now in the form \((y - k)^2 = 4a(x - h)\), where \(k = \frac{2}{3}\) and \(h = -\frac{61}{16}\). The axis of the parabola is given by the line \(y = k\):
\[ y = \frac{2}{3} \]

### Conclusion
Thus, the axis of the parabola is:
\[ 3y = 2 \]