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 its focus and vertex, we can follow these steps:

### Step 1: Identify the coordinates of the focus and vertex
- Focus (F) = (2, 3)
- Vertex (V) = (-1, 1)

### Step 2: Determine the midpoint between the focus and the directrix
The vertex of the parabola bisects the line segment joining the focus and the foot of the directrix (D). We can denote the coordinates of the foot of the directrix as (x, y).

Using the midpoint formula:
\[ V_x = \frac{F_x + D_x}{2} \quad \text{and} \quad V_y = \frac{F_y + D_y}{2} \]
Substituting the known values:
\[ -1 = \frac{2 + x}{2} \quad \text{and} \quad 1 = \frac{3 + y}{2} \]

### Step 3: Solve for x and y
From the first equation:
\[ -1 = \frac{2 + x}{2} \implies -2 = 2 + x \implies x = -4 \]

From the second equation:
\[ 1 = \frac{3 + y}{2} \implies 2 = 3 + y \implies y = -1 \]

Thus, the coordinates of the foot of the directrix (D) are (-4, -1).

### Step 4: Find the slope of the axis of the parabola
The slope (m) of the axis can be calculated using the coordinates of the focus and vertex:
\[ m = \frac{F_y - V_y}{F_x - V_x} = \frac{3 - 1}{2 - (-1)} = \frac{2}{3} \]

### Step 5: Determine the slope of the directrix
Since the directrix is perpendicular to the axis, the slope of the directrix (m_d) can be found using the relationship:
\[ m \cdot m_d = -1 \implies \frac{2}{3} \cdot m_d = -1 \implies m_d = -\frac{3}{2} \]

### Step 6: Use point-slope form to find the equation of the directrix
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting the coordinates of the foot of the directrix (-4, -1) and the slope (-3/2):
\[ y - (-1) = -\frac{3}{2}(x - (-4)) \]
This simplifies to:
\[ y + 1 = -\frac{3}{2}(x + 4) \]
\[ y + 1 = -\frac{3}{2}x - 6 \]
\[ y = -\frac{3}{2}x - 7 \]

### Step 7: Rearranging to standard form
To express this in standard form:
\[ \frac{3}{2}x + y + 7 = 0 \implies 3x + 2y + 14 = 0 \]

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