What is the equation of the quadratic graph with a focus of (3,1) and a directrix of y=5

Explanation:

Assume a general point (x,y) that belong to the curve

To get the equation of the parabola, we follow the given steps:

1- Find the distance between point (x,y) and the focus point:

The general point is (x,y) and the focus is (3,1)

Using the distance formula:

D = .................> equation I

2- Find the distance between the point (x,y) and the directrix:

Directrix is a horizontal line having the general equation y=c

We are given that the directrix is y = 5

Therefore, the distance between the point and the directrix will simply be:

|y-c| = |y-5| ...............> equation II

3- Equate equations I and II and solve for y:

a- Square both sides:

(x-3)² + (y-1)² = (y-5)²

b- Now, expand the brackets having y:

(x-3)² + y² - 2y + 1 = y² - 10y + 25

c- Finally, solve for y:

(x-3)² + 1 - 25 = -10y + 2y

(x-3)² - 24 = -8y

Attached is the graph for the parabola.

Hope this helps :)