If the focus is (1,-1) and the directrix is the line x+2y-9=0, the vertex of the parabola is

Options

A. (1,2)
B. (2,1)
C. (1,-2)
D. (2,-1)

Answer

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

### Step 1: Identify the Focus and Directrix
The focus of the parabola is given as \( S(1, -1) \) and the directrix is the line \( x + 2y - 9 = 0 \).

### Step 2: Find the Slope of the Directrix
The equation of the directrix can be rewritten in slope-intercept form:
\[ 2y = -x + 9 \implies y = -\frac{1}{2}x + \frac{9}{2} \]
From this, we can see that the slope \( m_1 \) of the directrix is \( -\frac{1}{2} \).

### Step 3: Determine the Slope of the Axis of the Parabola
The slope of the axis of the parabola is the negative reciprocal of the slope of the directrix. Therefore:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2 \]

### Step 4: Write the Equation of the Axis
Using the point-slope form of the line, we can write the equation of the axis which passes through the focus \( S(1, -1) \):
\[ y - (-1) = 2(x - 1) \]
This simplifies to:
\[ y + 1 = 2x - 2 \implies 2x - y - 3 = 0 \]

### Step 5: Find the Intersection Point of the Directrix and the Axis
To find the intersection point \( P \) of the directrix and the axis, we need to solve the two equations:
1. Directrix: \( x + 2y - 9 = 0 \)
2. Axis: \( 2x - y - 3 = 0 \)

From the second equation, we can express \( y \) in terms of \( x \):
\[ y = 2x - 3 \]
Substituting this into the directrix equation:
\[ x + 2(2x - 3) - 9 = 0 \]
This expands to:
\[ x + 4x - 6 - 9 = 0 \implies 5x - 15 = 0 \implies x = 3 \]
Now substituting \( x = 3 \) back to find \( y \):
\[ y = 2(3) - 3 = 6 - 3 = 3 \]
Thus, the coordinates of point \( P \) are \( (3, 3) \).

### Step 6: Find the Vertex \( Q \)
The vertex \( Q \) is the midpoint between the focus \( S(1, -1) \) and the intersection point \( P(3, 3) \). Using the midpoint formula:
\[ Q = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + 3}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{4}{2}, \frac{2}{2} \right) = (2, 1) \]

### Final Answer
The vertex of the parabola is \( Q(2, 1) \).

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