Find the equation of the parabola determined by the given information.

Focus (3,5), directrix x = -1

Solution:-

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 the parabola is x = -1, then -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 equation. 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 for h , p, and k into standard equation of a parabola.

$(y - 5)^{2}$ = 4*2(x – 1)

$(y - 5)^{2}$ = 8(x – 1)