Parabola: Formula, Focus, Directrix

Parabola: Formula, Focus, Directrix

How to solve parabola problems: definition, formula (equation, focus, directrix), proof, examples, and their solutions.

Definition

A parabola is the set of the points that are equidistant from the focus and the directrix.

A parabola is the set of the points
that are equidistant
from the focus and the directrix.

Formula 1

Parabola: y^2 = 4px. Focus: (p, 0), Directrix: x = -p.

Parabola: y2 = 4px

Focus: (p, 0)
Directrix: x = -p

Proof: Formula 1

Parabola: Proof of Formula 1

Set [distance between (x, y) and (p, 0)]
= [distance between (x, y) and x = -p].

Square both sides.

Left side
Square of a difference

Right side
Square of a sum

Cancel the gray terms on both sides.

Then y2 = 4px.

Example 1

Find the focus and the directrix of the given parabola. y^2 = 8x

y2 = 8x
= 4⋅2⋅x.

So p = 2.

Then the focus is (2, 0).
And the directrix is x = -2.

See how the graph of y2 = 4⋅2⋅x looks like.

The graph is determined by
its focus (2, 0)
and its directrix x = -2.

Example 2

Write and equation for the parabola described below. focus: (3, 4), directrix: x = -1

Roughly draw the given conditions.

Focus: (3, 4)
Directrix: x = -1

The distance between
the focus (3, 4) and the vertex
is p.

And the distance between
the vertex and the directrix x = -1
is also p.

The focus, the vertex, and the diretrix
form a segment
whose length is 2p.

So 2p = 3 - (-1) = 4.

p = 2

The focus is not (2, 0).

So the parabola is under a translation.

Translation of a point

So use p = 2
to find the changes of x and y:
(3, 4) = (2 + 1, 4).

So the parabola is under the translation
(x, y) → (x + 1, y + 4).

Write the equation of the parabola
using the above conditions.

(y - 4)2 = 4⋅2⋅(x - 1)

Translation of a function

See how (y - 4)2 = 8x = 4⋅2⋅(x - 1) looks like.

The graph is determined by
its focus (2 + 1, +4)
and its directrix x = -2 + 1.

Formula 2

Parabola: x^2 = 4py. Focus: (0, p), Directrix: y = -p.

Parabola: x2 = 4py

Focus: (0, p)
Directrix: y = -p

Proof: Formula 2

Parabola: Proof of Formula 2

Set [distance between (x, y) and (0, p)]
= [distance between (x, y) and y = -p].

Square both sides.

Left side
Square of a difference

Right side
Square of a sum

Cancel the gray terms on both sides.

Then x2 = 4py.

Example 3

Find the focus and the directrix of the given parabola. y = x^2

x2 = y
= 4⋅(1/4)⋅y.

So p = 1/4.

Then the focus is (0, 1/4).
And the directrix is y = -1/4.

See how the graph of x2 = 4⋅(1/4)⋅y looks like.

The graph is determined by
its focus (0, 1/4)
and its directrix y = -1/4.

Example 4

Find the focus and the directrix of the given parabola. 4y = x^2 - 2x + 9

Switch both sides
to set the x2 term on the left side.

Move the constant term +9 to the right side.

And complete the square on the left side.

Complete the square

Factor the right side.

The parabola is (x - 1)2 = 4⋅1⋅(y - 2).

This is the image of x2 = 4⋅1⋅y
under the translation (x, y) → (x + 1, y + 2).

Translation of a function

So the focus is (+1, 1 + 2) = (1, 3).
And the directrix is y = -1 + 2, y = 1.

See how the graph of (x - 1)2 = 4⋅1⋅(y - 2) looks like.

The graph is determined by
its focus (+1, 1 + 2)
and its directrix y = -1 + 2.