Find the position of the point (-2,2) with respect to the parabola `y^2-4y+9x+13=0`.

Answer

To determine the position of the point (-2, 2) with respect to the parabola given by the equation \( y^2 - 4y + 9x + 13 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation of the parabola
We start with the given equation:
\[ y^2 - 4y + 9x + 13 = 0 \]
We can rearrange this equation to isolate \( y \):
\[ y^2 - 4y = -9x - 13 \]

### Step 2: Complete the square for the \( y \) terms
Next, we complete the square for the \( y \) terms on the left side:
\[ y^2 - 4y + 4 = -9x - 13 + 4 \]
This simplifies to:
\[ (y - 2)^2 = -9x - 9 \]
or
\[ (y - 2)^2 = -9(x + 1) \]

### Step 3: Identify the vertex of the parabola
From the equation \( (y - 2)^2 = -9(x + 1) \), we can see that the vertex of the parabola is at the point \((-1, 2)\).

### Step 4: Substitute the point (-2, 2) into the equation
Now, we will substitute the point (-2, 2) into the equation to check its position:
\[ (y - 2)^2 + 9(x + 1) = 0 \]
Substituting \( x = -2 \) and \( y = 2 \):
\[ (2 - 2)^2 + 9(-2 + 1) = 0 \]
This simplifies to:
\[ 0 + 9(-1) = 0 \]
or
\[ 0 - 9 = -9 \]

### Step 5: Analyze the result
Since the result is \(-9\), which is less than \(0\), this indicates that the point (-2, 2) lies inside the parabola.

### Conclusion
Thus, the position of the point (-2, 2) with respect to the parabola \( y^2 - 4y + 9x + 13 = 0 \) is that it lies **inside** the parabola.

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