Parabola (Focus-Directrix Definition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parabola (Focus-Directrix Definition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every point on the curve is the same distance from the focus as from the directrix line.

Common stuck point: The procedure for parabola (focus-directrix definition) is the easy part; the trap is equating $p$ with the coefficient $a$ . Asking "Is the curve the set of points equally far from a single point and a single line, with only one variable squared?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the curve the set of points equally far from a single point and a single line, with only one variable squared?

Worked Examples

Example 1

easy

Find the focus and directrix of the parabola

y = \frac{1}{8}x^2

Answer

\text{Focus: } (0, 2); \quad \text{Directrix: } y = -2

First step

Rewrite in standard form:

x^2 = 8y

. This matches

x^2 = 4py

where

4p = 8

, so

p = 2

Full solution

2
The parabola opens upward. The focus is at $(0, p) = (0, 2)$ .
3
The directrix is $y = -p = -2$ .

For a parabola

x^2 = 4py

, the parameter

p

is the distance from the vertex to the focus (and also from the vertex to the directrix). If

p > 0

, the parabola opens upward; if

p < 0

, it opens downward.

Example 2

medium

Write the equation of a parabola with vertex at the origin, opening to the right, with focus at

(3, 0)

Example 3

medium

Find the focus and directrix of the parabola

y = \frac{1}{8}x^2

Example 4

medium

Find the equation of the parabola with focus

(0, 5)

and directrix

y = -5

Example 5

medium

Find the vertex, focus, and directrix of

y^2 - 4y - 8x + 4 = 0

Example 6

medium

Show that the point

(4, 4)

lies on the parabola with focus

(0, 1)

and directrix

y = -1

by checking the equidistance property.

Example 7

medium

A parabolic mirror has its vertex at the origin and a depth of

4

cm at a width of

8

cm. Where should a bulb be placed for parallel reflected rays?

Example 8

hard

Find the equation of the parabola with focus

(3, 2)

and directrix

x = -1

Example 9

hard

Derive the equation of the parabola whose points are equidistant from the focus

(0, 2)

and directrix

y = -2

Example 10

hard

A satellite dish has a parabolic cross-section. The dish is

1.2

m across at the rim and

0.2

m deep at the center. Where (in meters from the vertex along the axis) should the receiver go?

Example 11

challenge

A chord of the parabola

y^2 = 4x

passes through the focus and has length

9

. Find the equation of the chord (assume it is not the latus rectum).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Find the equation of the parabola with focus

(0, -4)

and directrix

y = 4

Example 2

hard

A parabola has vertex

(2, -1)

and focus

(2, 3)

. Find its equation and the length of the latus rectum.

Example 3

easy

A parabola has vertex at the origin and focus

(0,2)

. Where is the directrix?

Example 4

easy

For

x^2 = 8y

, find

p

Example 5

easy

For

x^2 = 8y

, find the focus.

Example 6

easy

For

x^2 = 8y

, give the directrix.

Example 7

easy

Which way does

y^2 = 12x

open?

Example 8

easy

Write the equation of a parabola with vertex origin, opening up,

p=3

Example 9

easy

For

y^2 = 12x

, find

p

Example 10

easy

For

y^2 = 12x

, find the focus.

Example 11

medium

Find the focus and directrix of

y = \frac{1}{8}x^2

Example 12

medium

Find

p

for

y = \frac{1}{4}x^2

Example 13

medium

Find the equation of a parabola with focus

(0,3)

and directrix

y=-3

Example 14

medium

Find the focus of

y^2 = -8x

Example 15

medium

Find the equation of a parabola with vertex

(2,1)

, focus

(2,4)

Example 16

medium

Find

p

for the parabola

(x-1)^2 = 16(y+2)

Example 17

medium

A point

(x,y)

is equidistant from focus

(0,1)

and directrix

y=-1

. Find the equation.

Example 18

challenge

Find the equation of a parabola with focus

(3,0)

and directrix

x=-3

Example 19

challenge

The latus rectum (focal chord through focus, perpendicular to axis) of

x^2=4py

has length

|4p|

. Find it for

x^2=8y

Example 20

challenge

Find the vertex and focus of

y = x^2 - 4x + 7

Example 21

medium

Find the directrix of

y^2 = -8x

Example 22

medium

Find the focus of

(y-1)^2 = 8(x-2)

Example 23

easy

For

x^2 = 16y

, find the value of

p

Example 24

easy

Which direction does the parabola

y^2 = -8x

open?

Example 25

easy

Write the equation of a parabola with vertex at the origin, opening down, with

p = 2

Example 26

medium

Find the focus and directrix of

(x - 1)^2 = 12(y + 2)

Example 27

medium

A parabola has vertex

(3, 1)

and directrix

y = 4

. Find its equation.

Example 28

medium

Find the equation of the parabola with focus

(2, 3)

and directrix

x = -4

Example 29

medium

Find the equation of the parabola with vertex

(0, 0)

that passes through

(2, 4)

and opens upward.

Example 30

hard

For the parabola

(x - 2)^2 = 16(y + 1)

, find the equation of the latus rectum (the focal chord perpendicular to the axis).

Example 31

hard

A parabola has equation

y = 2x^2 - 8x + 7

. Find its vertex, focus, and directrix.

Example 32

hard

Find the equation of the parabola with vertex at the origin, axis along the

x

-axis, passing through

(2, 4)

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Related Concepts

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Background Knowledge

These ideas may be useful before you work through the harder examples.

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