If a parabola has a vertex of (1,2) and a directrix of y=2.5, find the coordinate of the parabola's focus. (15 points)

The focus is a point and the directrix is a line.
The parabola has the equation as follows:

(x - h)² = 4p(y - k)
 
where (h,k) is the vertex, p is the distance from the vertex to the focus and also the distance from the vertex to the focus.  The focus is (h, k + p) and the directrix is y = k - p

We can now write the equation.

(x - h)² = 4p(y - k) 
(x - 1)² =  4p(y - 2)

y= 2.5 = k-p = 2 - p
p = -0.5

(x - 1)² =  4(-0.5)(y - 2)
(x - 1)² =  -2 (y - 2)

The focus would be 
(1 , 2 + -0.5)
(1 , 1.5 )

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