Question

Use the distance formula to find the equation of a parabola. Focus: (6,0) and directrix X = -3

I know the distance formula is:
√(x-x1)^2+(y-y1)^2 = √(x-x2)^2+(y-y2)^2

I know I need to sub (6,0) for x1,y1 and (-3,y) for x2,y2

√(x-6)^2+(y-0)^2 = √(x-(-3))^2+(y-y)^2
√(x-6)^2+y^2 = √(x+3)^2

Then I get lost. Please help.

Answer 1

focus(6,0), directrix x=-3 means

If the distances are equal, then the squares of the distances from directrix and focus are also equal, so
(x-(-3))²=(x-6)^2+(y-0)^2

(x+3)²-(x-6)²=y²
Factor by difference of two squares
18(x-1.5)=y²
(x-1.5)=y²/(4*4.5)
h=1.5, c=4.5 for (x-h)=y²/4c
meaning the vertex is at (1.5,0) and distance from vertex to directrix is 4.5.