How do you find the standard equation given focus (8,10), and vertex (8,6)?
The focus is above the vertex, therefore, the vertex form of the equation is:
Use the focus to compute
Expand the equation into standard form.
The focal distance, f, is the distance form the vertex to the focus:
Compute the value of "a":
The vertex tells us that
Substituting these values into the vertex form:
Expand the square:
Find the distance between vertex and focus. Call that p.
Since it opens upward, p >0. Use (x-h)2 = 4p(y - k).
The equation of the parabola that opens up or down and has vertex (h, k) is
where p is the difference between the y-coordinates of the focus and the vertex.
In this example, p = 10 - 6 = 4, and (h, k) = (8, 6). Therefore,
This may have been the form you were seeking.
[If we want this in the "standard form," that usually means solving for the variable that is not squared. Distribute the 16, and add...
If you were only interested in the standard form, set a = 1/(4p) and go straight to the vertex form:
a = 1/(4*4) = 1/16. Therefore,
Use FOIL and distribute if you prefer the form
I will not spoil that fun for you.