College Algebra & Trigonometry, 2018a

5.2. THE EQUATION OF THE PARABOLA 267
We can derive the standard equation for a parabola using the distance formula.
d
(x, y)
Focus: (h, k + p)
d
Vertex: (h, k)
Directrix: y = k − p
In the picture above the two distances labeled “d” should be the same distance.
The vertical distance between the point (x, y) and the directrix y = k − p is simply
the difference between their y coordinates:
d = y − (k − p)
To find the distance between the point (x, y) and the focus (h, k + p) we need to
use the distance formula:
d = √ (x − h) 2 +(y − (k + p)) 2
Then we set the two distances equal to each other:
√
(x − h)2 +(y − (k + p)) 2 = y − (k − p)
Square both sides:
(x − h) 2 +(y − (k + p)) 2 =(y − (k − p)) 2
We’ll need to expand each side and collect like terms, but we’ll leave the (x − h) 2
alone because it will appear in this form in the final equation.
(x − h) 2 + y 2 − 2(k + p)y +(k + p) 2 = y 2 − 2(k − p)y +(k − p) 2