If (3, 6) is vertex and (4, 5) is focus of parabola then equation of directrix is

Options

A. x + y + 5 = 0
B. x + y - 5 = 0
C. X - y - 5 = 0
D. X - y + 5 = 0

Answer

To find the equation of the directrix of the parabola given the vertex and focus, we can follow these steps:

### Step 1: Identify the Vertex and Focus
The vertex \( V \) of the parabola is given as \( (3, 6) \) and the focus \( F \) is given as \( (4, 5) \).

### Step 2: Find the Coordinates of the Directrix
Let the coordinates of the directrix be \( D(h, k) \). According to the properties of a parabola, the vertex \( V \) is the midpoint between the focus \( F \) and the directrix \( D \).

Using the midpoint formula:
\[ V = \left( \frac{h + 4}{2}, \frac{k + 5}{2} \right) \]
Since \( V = (3, 6) \), we can set up the following equations:
1. \( \frac{h + 4}{2} = 3 \)
2. \( \frac{k + 5}{2} = 6 \)

### Step 3: Solve for \( h \) and \( k \)
From the first equation:
\[ h + 4 = 6 \implies h = 2 \]

From the second equation:
\[ k + 5 = 12 \implies k = 7 \]

Thus, the coordinates of the point where the directrix intersects the axis of the parabola are \( D(2, 7) \).

### Step 4: Determine the Slope of the Axis
The slope of the axis of the parabola can be calculated using the coordinates of the focus and vertex. The slope \( m \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 6}{4 - 3} = \frac{-1}{1} = -1 \]

### Step 5: Find the Slope of the Directrix
Since the axis and the directrix are perpendicular, the product of their slopes is -1. Therefore, if the slope of the axis is -1, the slope of the directrix \( m_d \) is:
\[ m_d \cdot (-1) = -1 \implies m_d = 1 \]

### Step 6: Write the Equation of the Directrix
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (2, 7) \) and \( m = 1 \):
\[ y - 7 = 1(x - 2) \]
This simplifies to:
\[ y - 7 = x - 2 \implies y = x + 5 \]

### Step 7: Rearranging to Standard Form
Rearranging the equation gives:
\[ x - y + 5 = 0 \]

Thus, the equation of the directrix is:
\[ \boxed{x - y + 5 = 0} \]