Find the equation for the parabola with focus (3,4) and vertex (3,3)?

Question

How do I solve this?

Answer 1

You can the definition of parabola to get the answer.

First find the dirextrix which has an equal distance to the vertex as the focus. So, the directrix is

y = 2

Now use the definition that any point (x, y) on a parabola is equil - distant from the focus and the dirextrix.

Therefore, you have

(x-x)^2+(y-2)^2 = (x-3)^2 + (y-4)^2

Simplify,

4y = (x-3)^2 + 12

y = (1/4)(x-3)^2 + 3

Answer 2

First of all, you need to determine if this parabola is opening vertically or horizontally.  Since the vertex and the focus share the same x-value, the line of symmetry is at x = 3, which is vertical.  The standard form of a vertical parabola is.

4p*(y-k) = (x - h)^2, where (h,k) are the coordinates of the vertex, and p is the distance from the vertex to the focus.

We have (h,k) = (3,3) and p = 4 - 3 = 1

4*1*(y-3) = (x-3)^2

y - 3 = 1/4(x-3)^2

y = 1/4(x-3)^2 + 3