Find the equation if 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 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)