Parabolas: focus and directrix

Warm-up

Three quick questions from earlier lessons.

Discover

A satellite dish is a parabola: every signal coming straight in bounces to one point, the focus. For $x^2 = 8y$ , where is it?

$(0, 2)$
$(0, 8)$

Key idea

A parabola is the set of points as far from a point (the focus) as from a line (the directrix).

x^2 = 4py

: vertex

(0, 0)

, focus

(0, p)

, directrix

y = -p

, opens up if

p > 0

.

y^2 = 4px

: focus

(p, 0)

, directrix

x = -p

, opens right if

p > 0

.

Worked example

Find the focus and directrix of $y^2 = -12x$ .

4p = -12

, so

p = -3

Match with y² = 4px.

Focus

(-3, 0)

: it opens to the left

Focus (p, 0).

Directrix

x = 3

Directrix x = −p.

Your turn, guided

Your turn: $x^2 = 20y$ .

p = 5

, focus

(0, 5)

4p = 20.

Directrix

y = -5

y = −p: type the number.

Practice ★

The parabola $x^2 = 8y$ has its focus at $(0, c)$ . What is $c$ ?
The parabola $x^2 = 24y$ has its focus at $(0, c)$ . What is $c$ ?
The parabola $x^2 = 4y$ has its focus at $(0, c)$ . What is $c$ ?

Watch out!

Practice ★★

The parabola $y^2 = 8x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 20x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 4x$ has the directrix $x = d$ . What is $d$ ?

A parabola has focus $(0, 3)$ and directrix $y = -3$ . Its equation is $x^2 = ky$ . What is $k$ ?
A parabola has focus $(0, 2)$ and directrix $y = -2$ . Its equation is $x^2 = ky$ . What is $k$ ?

Exit ticket

Three questions to check you have got it.

The parabola $y^2 = 8x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 20x$ has the directrix $x = d$ . What is $d$ ?
The parabola $y^2 = 4x$ has the directrix $x = d$ . What is $d$ ?

Summary card

Parabolas

$x^2 = 4py$ : focus $(0, p)$ , directrix $y = -p$ .
$y^2 = 4px$ : focus $(p, 0)$ , directrix $x = -p$ .

Watch out: divide by 4 to get $p$ .

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