To find the equation of a parabola with a given focus and directrix, we typically use the definition of a parabola: it is the set of all points (x, y) that are equidistant from the focus and the directrix.
Here, we are given the focus \( F(6, -5) \) and the directrix \( x = 2 \).
Step 1: Identify the coordinates of the focus
Focus: \( F(6, -5) \)
Step 2: Determine the position of the directrix
The directrix, being a vertical line, has the equation \( x = 2 \).
Step 3: Identify the vertex of the parabola
The vertex of the parabola lies halfway between the focus and the directrix. To find the x-coordinate of the vertex \( V \):
\[ V_x = \frac{F_x + d_x}{2} = \frac{6 + 2}{2} = \frac{8}{2} = 4 \]
The y-coordinate of the vertex is the same as the y-coordinate of the focus since the directrix is vertical:
\[ V_y = -5 \]
So, the vertex \( V \) is at \( (4, -5) \).
Step 4: Determine the distance from the vertex to the focus (p)
The distance \( p \) is the distance from the vertex to the focus along the x-direction since the parabola opens to the right (the focus is to the right of the vertex):
\[ p = F_x - V_x = 6 - 4 = 2 \]
Step 5: Write the standard form of the parabola's equation
For a parabola that opens to the right with vertex \( (h, k) \) at \( (4, -5) \) and distance \( p \) from the vertex:
The standard form of the equation is:
\[ (y - k)^2 = 4p(x - h) \]
Substituting \( h = 4 \), \( k = -5 \), and \( p = 2 \):
\[ (y + 5)^2 = 4 \cdot 2 (x - 4) \]
\[ (y + 5)^2 = 8(x - 4) \]
Final Equation
Thus, the equation of the parabola is:
\[ (y + 5)^2 = 8(x - 4) \]