Find the equation if the parabola determined by the given information.
Focus (3,5), directrix x = -1.
Notice that the directrix is a vertical line. Since the axis of symmetry is perpendicular to the directrix, then it is a horizontal line.
The parabola with a horizontal axis of symmetry will have directrix x = h – p, focus (h +p, k), and standard equation (y – k)2 = 4p(x – h).
Since the directrix of this parabola is x = -1, then – 1 = h – p.
Since the x – value of the focus is 3, then 3 = h + p.
Find h and p using any method for solving a system of equations. Using the elimination method, eliminate p and solve for h.
h = 1
Substitute the value for h into one of the equation to solve for p.
p = 2
k is the y-value of the focus. So, k = 5.
Substitute the values for h, p, and k into the standard equation of a parabola.
(y – 5)2 = 4 *2(x – 1)
(y – 5)2 = 8(x – 1)