Use the distance formula to write an equation of the parabola. For these following problems:

Question

1. Focus: (0,1) Directrix: y = -1
2. Focus: (0,-2) Directrix: y = 2
3. Vertex: (0,0) Directrix: y = -6
4. Vertex: (0,0) Focus: (0,-10)

Answer 1

You have the same problem 4 times. You should only ask it once, and learn from the solution how to solve the other 3.

I will solve 1 and show you how to get the focus on 3.

The distance of a point (x, y) from the focus (0, 1) is sqrt((x-0)² + (y-1)²) = sqrt(x² + (y-1)²)

The distance of a point (x,y) from y = -1 is sqrt((y--1)²) = sqrt((y+1)²)

Then, we will set the two distances, or rather their squares, equal.

x² + (y-1)²= (y+1)²

x² + y² - 2y + 1= y² + 2y + 1

x² = 4y

If you prefer, y = x²/4, but the former is the way it is frequently written, as it allows one to immediately see that the focus is one above the vertex, and the directrix is one below.

For 3, we can determine that the focus is opposite the directrix from the vertex. Thus,

the focus is at (0, 0 - -10) = (0, 10)