Find the equation of the dirctly of the parabola whose focus and vertex are (5,3) and (3,1) respectively .

Options

A. `x+ y +1 = 0 `
B. `x - y = 0 `
C. `x+y = 0 `
D. `x- y - 1 = 0 `

Answer

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

### Step 1: Identify the Focus and Vertex
The focus \( S \) is given as \( (5, 3) \) and the vertex \( A \) is given as \( (3, 1) \).

### Step 2: Find the Midpoint
Let \( Z \) be the projection of the focus \( S \) on the directrix. Since \( A \) is the midpoint of \( S \) and \( Z \), we can use the midpoint formula:
\[ A = \left( \frac{x_S + x_Z}{2}, \frac{y_S + y_Z}{2} \right) \]
Substituting the coordinates of \( A \) and \( S \):
\[ (3, 1) = \left( \frac{5 + x_Z}{2}, \frac{3 + y_Z}{2} \right) \]

### Step 3: Solve for \( x_Z \) and \( y_Z \)
From the x-coordinate:
\[ 3 = \frac{5 + x_Z}{2} \]
Multiplying both sides by 2:
\[ 6 = 5 + x_Z \implies x_Z = 1 \]

From the y-coordinate:
\[ 1 = \frac{3 + y_Z}{2} \]
Multiplying both sides by 2:
\[ 2 = 3 + y_Z \implies y_Z = -1 \]

Thus, the coordinates of point \( Z \) are \( (1, -1) \).

### Step 4: Find the Slope of Line \( SA \)
Now, we need to find the slope of the line segment \( SA \):
\[ \text{slope of } SA = \frac{y_A - y_S}{x_A - x_S} = \frac{1 - 3}{3 - 5} = \frac{-2}{-2} = 1 \]

### Step 5: Find the Slope of the Directrix
The slope of the directrix is perpendicular to the slope of \( SA \). Therefore, if the slope of \( SA \) is \( 1 \), the slope of the directrix will be:
\[ \text{slope of directrix} = -\frac{1}{\text{slope of } SA} = -1 \]

### Step 6: Write the Equation of the Directrix
Using the point-slope form of the line equation, where the point is \( (1, -1) \) and the slope is \( -1 \):
\[ y - (-1) = -1(x - 1) \]
This simplifies to:
\[ y + 1 = -x + 1 \]
Rearranging gives:
\[ x + y + 1 = 0 \]

### Final Answer
The equation of the directrix is:
\[ x + y + 1 = 0 \]