Let's solve the problem step by step using the given details:
Step 1: Analyze the given elements
We have the focus of the parabola [tex]\( F(3, 2) \)[/tex] and the directrix equation [tex]\( x + 1 = 0 \)[/tex].
Step 2: Convert the directrix to a usable form
The directrix [tex]\( x + 1 = 0 \)[/tex] can be rewritten as [tex]\( x = -1 \)[/tex]. This represents a vertical line located at [tex]\( x = -1 \)[/tex].
Step 3: Determine the vertex of the parabola
The vertex of the parabola is exactly halfway between the focus and the directrix line along the x-axis.
To find the x-coordinate of the vertex, we can take the average of the x-coordinates of the focus and the directrix:
[tex]\[ x_{\text{vertex}} = \frac{x_F + x_{\text{directrix}}}{2} \][/tex]
[tex]\[ x_{\text{vertex}} = \frac{3 + (-1)}{2} = \frac{2}{2} = 1 \][/tex]
The y-coordinate of the vertex is the same as the y-coordinate of the focus, which is [tex]\( 2 \)[/tex]. Therefore, the vertex is at [tex]\( (1, 2) \)[/tex].
Step 4: Determine the parameter [tex]\( p \)[/tex]
The parameter [tex]\( p \)[/tex] represents the distance between the vertex and the focus. This distance is measured along the x-axis since the focus lies on a horizontal line through the vertex.
We can calculate [tex]\( p \)[/tex] as follows:
[tex]\[ p = x_F - x_{\text{vertex}} \][/tex]
[tex]\[ p = 3 - 1 = 2 \][/tex]
Summary
Thus, the vertex of the parabola is [tex]\( (1, 2) \)[/tex] and the parameter [tex]\( p \)[/tex] is [tex]\( 2 \)[/tex].
Final answer:
- Vertex: [tex]\( \mathbf{(1, 2)} \)[/tex]
- Parameter [tex]\( p \)[/tex]: [tex]\( \mathbf{2} \)[/tex]
