If the focus and the vertex of a parabola are (2,3) and (-1,1) respectively, then the directrix is :

Options

A. `3x+2y+14=0`
B. `3x+2y-25=0`
C. `2x-3y+10=0`
D. `2x-3y+14=0`

Answer

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

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

### Step 2: Use the Midpoint Formula
The vertex \( V \) is the midpoint between the focus \( F \) and the directrix. Let the coordinates of the directrix be \( D(\alpha, \beta) \). According to the midpoint formula:
\[ V = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This gives us two equations:
1. For the x-coordinates:
\[ \frac{2 + \alpha}{2} = -1 \]
2. For the y-coordinates:
\[ \frac{3 + \beta}{2} = 1 \]

### Step 3: Solve for \( \alpha \)
From the first equation:
\[ \frac{2 + \alpha}{2} = -1 \]
Multiplying both sides by 2:
\[ 2 + \alpha = -2 \]
Subtracting 2 from both sides:
\[ \alpha = -4 \]

### Step 4: Solve for \( \beta \)
From the second equation:
\[ \frac{3 + \beta}{2} = 1 \]
Multiplying both sides by 2:
\[ 3 + \beta = 2 \]
Subtracting 3 from both sides:
\[ \beta = -1 \]

### Step 5: Find the Equation of the Directrix
Now we have the coordinates of the directrix as \( D(-4, -1) \).

### Step 6: Determine the Slope of the Line
Next, we need to find the slope of the line connecting the focus and vertex. The slope \( m_1 \) is given by:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - (-1)} = \frac{2}{3} \]

### Step 7: Find the Perpendicular Slope
Since the directrix is perpendicular to the line connecting the focus and vertex, the slope \( m_2 \) of the directrix is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \]

### Step 8: Write the Equation of the Directrix
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( D(-4, -1) \) and \( m_2 = -\frac{3}{2} \):
\[ y - (-1) = -\frac{3}{2}(x - (-4)) \]
This simplifies to:
\[ y + 1 = -\frac{3}{2}(x + 4) \]
Distributing:
\[ y + 1 = -\frac{3}{2}x - 6 \]
Rearranging gives:
\[ \frac{3}{2}x + y + 7 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 3x + 2y + 14 = 0 \]

### Final Answer
The equation of the directrix is:
\[ 3x + 2y + 14 = 0 \]