Parabola: Equation
How to use a parabola equation to find the focus and the directrix (and vice versa): definition, formula, 6 examples, and their solutions.
Definition
A parabola is the set of the points
that are equidistant
from the focus (blue point)
and the directrix (dashed line).
Formulay2 = 4px
The black curve is the graph of a parabola
y2 = 4px.
For y2 = 4px,
the focus is (p, 0)
and the directrix is x = -p.
ExampleFocus
Change the parabola to y2 = 4px form.
y2 = 4⋅2⋅x
y2 = 4⋅2⋅x
Then the focus is
(2, 0).
So
(2, 0)
is the answer.
ExampleDirectrix
You just found that
the given equation y2 = 8x is
y2 = 4⋅2⋅x.
y2 = 4⋅2⋅x
Then the directrix is
x = -2.
So
x = -2
is the answer.
ExampleParabola Equation
The focus is (3, 0).
And the directrix is x = -3.
Then the equation of the parabola is
y2 = 4⋅3⋅x.
4⋅3⋅x = 12x
So
y2 = 12x
is the answer.
Formulax2 = 4py
For the parabola x2 = 4py,
the focus is (0, p)
and the directrix is y = -p.
ExampleFocus
Change the parabola to x2 = 4py form.
Switch both sides.
y = 4⋅[1/4]⋅y
x2 = 4⋅[1/4]⋅y
Then the focus is
(0, 1/4).
So
(0, 1/4)
is the answer.
ExampleDirectrix
You just found that
the given equation y = x2 is
x2 = 4⋅[1/4]⋅y.
x2 = 4⋅[1/4]⋅y
Then the directrix is
y = -1/4.
So
y = -1/4
is the answer.
ExampleParabola Equation
The focus is (0, 2).
And the directrix is y = -2.
Then the equation of the parabola is
x2 = 4⋅2⋅y.
4⋅2⋅y = 8y
So
x2 = 8y
is the answer.