Parabola Question

Question

Find the focus, directrix, focal diameter, vertex and axis of symmetry for the parabola:

40.64y=x^2

Answer 1

64y = x^2

y = (1/64)x^2 has vertex at the origin, (0,0). the origin is also the minimum point of the parabola. the origin is also the x and y intercepts. axis of symmetry is x=0.

focus is on the y axis above the origin. directrix is equidistant to the origin, below the origin.

x coordinate of the focus is 0. (0,f) go to the right to the parabola curve, at point (x,f). from (0,f) to (x,f) is a distance of x. from (x,f) to the directrix is a distance of x. f = (1/64)x^2, f=x/2, x/2 = (1/64)x^2, 1/2 = (1/64)x, x = 32

the focus is (0,16) the directrix is y=-16

the point (32,16) is on the parabola. 64(16) = (32)^2

It's a distance 32 from the focus and a distance 32 from the directrix

focus: (0,16)

directrix: y=-16

vertex: (0,0)

line of symmetry: x=0