Find the equation of the directrix of the parabola whose vertix is at (3,-2) and focus is at (6,2).

Answer

To find the equation of the directrix of the parabola with a vertex at (3, -2) and a focus at (6, 2), we can follow these steps:

### Step 1: Identify the Vertex and Focus
The vertex \( V \) of the parabola is given as \( (3, -2) \) and the focus \( S \) is given as \( (6, 2) \).

### Step 2: Calculate the Slope of the Axis
The slope of the axis of the parabola can be calculated using the formula for the slope between two points:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the focus and vertex:
\[ m_1 = \frac{2 - (-2)}{6 - 3} = \frac{4}{3} \]

### Step 3: Determine the Slope of the Directrix
The slope of the directrix \( m_2 \) is perpendicular to the slope of the axis. For two perpendicular lines, the product of their slopes is -1:
\[ m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1} = -\frac{3}{4} \]

### Step 4: Find the Coordinates of Point P
Since the vertex \( V \) is the midpoint of the line segment \( PS \) (where \( P \) is a point on the directrix), we can use the midpoint formula:
\[ V = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Let \( P = (x, y) \) and \( S = (6, 2) \). Thus, we have:
\[ 3 = \frac{6 + x}{2} \quad \text{and} \quad -2 = \frac{2 + y}{2} \]

### Step 5: Solve for x and y
From the first equation:
\[ 6 + x = 6 \implies x = 0 \]
From the second equation:
\[ 2 + y = -4 \implies y = -6 \]
So, the coordinates of point \( P \) are \( (0, -6) \).

### Step 6: Write the Equation of the Directrix
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \( P(0, -6) \) and \( m_2 = -\frac{3}{4} \):
\[ y - (-6) = -\frac{3}{4}(x - 0) \]
This simplifies to:
\[ y + 6 = -\frac{3}{4}x \]
Rearranging gives:
\[ \frac{3}{4}x + y + 6 = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 3x + 4y + 24 = 0 \]

### Final Answer
The equation of the directrix is:
\[ 3x + 4y + 24 = 0 \]