The equation of a parabola, step by step
A circle is the set of points a fixed distance from a centre.
The question
Write the equation of the parabola with focus (1, −5) and directrix y = 1.
The equation of a parabola
vertex = (1, −2)
p = −5 − (−2) = −3
(x − 1)² = −12(y + 2)
- Every point on a parabola is the same distance from the focus as from the directrix, so the vertex sits halfway between them: halfway from −5 to 1 is −2. The vertex is (1, −2).
- p is the gap from the vertex to the focus, and it is SIGNED: the focus is below the vertex, so p = −3 and the parabola opens downwards.
- Put the vertex and p into (x − h)² = 4p(y − k). With 4p = −12, that is the equation.
How it works.
- 01
Every point on a parabola is the same distance from the focus as from the directrix, so the vertex sits halfway between them: halfway from −5 to 1 is −2. The vertex is (1, −2).
- 02
p is the gap from the vertex to the focus, and it is SIGNED: the focus is below the vertex, so p = −3 and the parabola opens downwards.
- 03
Put the vertex and p into (x − h)² = 4p(y − k). With 4p = −12, that is the equation.