College algebra word problem

Question

Find an equation of a parabola with a vertex at the origin and the focus located at (x, y) = (4, 0).

Answer 1

because the focus is to the right of the vertex, it's a righward opening parabola of the general form

x-h = a(y-k)^2 (h,k) = (0,0) so

x = ay^2. all we need now is the value of a

the focus is equidistant to the origin with the directrix line. the directrix line is x=-4

the point on the parabola directly above the focus is (4,y). (4,y) is a distance y from the focus and also a distance y from the directrix line, which is a distance of 8 away. so y=8. the point (4,8) is on the parabola

plug it into x=ay^2 to get the value of a

4 = a(8)^2

4=64a

a = 4/64 = 1/16

the parabola equation is

x = (1/16)y^2 or

16x = y^2 or

y= +4sqr(x) or

y = +4x^1/2