The Parabola

Try this

Drag $P$ along the curve. Which is longer: $∣ P F ∣$ (solid) or $∣ P M ∣$ (dashed, the distance to the directrix)?
Where is the origin? How far is it from $F$ , and from the directrix $l$ ?
Make $p$ larger and smaller. How does the opening change? How does the coefficient in the equation change?
Drag $P$ to the point directly above $F$ (so that $P F$ is vertical). How many times $p$ is $∣ P F ∣$ now?

The locus of points in the plane that are equally far from a fixed point $F$ and a fixed line $l$ ( $F$ not on $l$ ) is called a parabola. $F$ is the focus, $l$ is the directrix, and $p$ is the distance from the focus to the directrix:

∣ P F ∣ = ∣ P M ∣

Predict first

A point $P$ on the parabola $y^{2} = 8 x$ has $x$ -coordinate $3$ . How far is $P$ from the focus?

Choose an answer first; then you can see the result.

Why y² = 2px

From the definition to the standard equation

Step 1 / 5

Choose the coordinates
Take the perpendicular from $F$ to $l$ as the $x$ -axis, and the midpoint of $F$ and the foot $K$ as the origin. Then $F (\frac{p}{2}, 0)$ and the directrix is $x = - \frac{p}{2}$ . In the figure $p = 2$ .

$F (\frac{p}{2}, 0), l : x = - \frac{p}{2}$
Distance to the focus
Distance to the directrix
Set them equal and square
The latus rectum

Parabolas opening to the left, upward or downward just swap $x$ and $y$ or change a sign: $y^{2} = - 2 p x$ , $x^{2} = 2 p y$ , $x^{2} = - 2 p y$ ( $p > 0$ ). The letter of the linear term names the axis, and the sign of the linear term gives the direction of the opening.

Three curves, one definition

The parabola says $∣ P F ∣ : ∣ P M ∣ = 1$ . What if you replace this ratio by any positive number $e$ ?

The focus and the directrix stay put; drag only the eccentricity

e

. What happens to the curve just before and just after

e = 1

Predict first

With the focus and the directrix fixed, slowly increase $e$ from $0.6$ to $1.4$ . What does the curve become, in order?

ellipse → parabola → hyperbolahyperbola → parabola → ellipsealways a parabola, just opening wider

Choose an answer first; then you can see the result.

Practice

Exercise 1Basic

Find the $x$ -coordinate of the focus of the parabola $y^{2} = 12 x$ .

Exercise 2Basic

The directrix of the parabola $x^{2} = - 8 y$ is $y = k$ . Find $k$ .

Exercise 3Advanced

$F$ is the focus of the parabola $y^{2} = 4 x$ , $O$ is the origin, and the point $P$ on the parabola has $∣ P F ∣ = 5$ . Find the area of $△ O P F$ .

Exercise 4Challenge

A line through the focus of the parabola $y^{2} = 4 x$ meets the parabola at $A$ and $B$ . The midpoint of $A B$ has $x$ -coordinate $3$ . Find $∣ A B ∣$ .

Try this

Why y² = 2px

From the definition to the standard equation

Three curves, one definition

Practice

Mistake Museum · Exhibits from this stop

Reading off the focus without the standard form

Taking $p$ as the $x$-coordinate of the focus