Parabolas, calculus lesson

A satellite dish is shaped so that every signal arriving straight on bounces off the surface and meets at a single point, where the receiver sits. Car headlights work the same way in reverse. That shape is a parabola, and the special point is its focus.

Key terms

Focus and directrix

Definition Let $F$ be a point and $L$ a line in the plane with $F$ not on $L$ . The parabola with focus $F$ and directrix $L$ is the set of all points $P$ that are equally far from $F$ and from $L$ ,

\lvert PF \rvert = \operatorname{dist}(P, L).

This is the focus directrix definition of a conic with eccentricity $e = 1$ .

In plain words Stand anywhere on the parabola. Your distance to the focus equals your distance to the directrix.

Example With $F = (0, 2)$ and $L$ the line $y = -2$ , the point $(4, 2)$ is on the parabola, since its distance to $F$ is $4$ and its distance to $L$ is $2 - (-2) = 4$ .

Vertex and axis

Definition The axis of a parabola is the line through the focus perpendicular to the directrix. The vertex is the point where the parabola meets its axis. It lies halfway between the focus and the directrix, and it is the point of the parabola closest to the focus.

In plain words The axis is the line of symmetry, and the vertex is the tip of the curve.

Example For focus $(0, 2)$ and directrix $y = -2$ , the axis is the $y$ axis and the vertex is $(0, 0)$ , the midpoint between $(0, 2)$ and $(0, -2)$ .

Standard equations

Definition Let $p \ne 0$ . The parabola with focus $(0, p)$ and directrix $y = -p$ has equation

x^2 = 4py.

It opens upward if $p > 0$ and downward if $p < 0$ . Likewise the parabola with focus $(p, 0)$ and directrix $x = -p$ is $y^2 = 4px$ , opening right if $p > 0$ and left if $p < 0$ . In both cases the vertex is the origin and $\lvert p \rvert$ is the distance from the vertex to the focus.

In plain words The equal distance condition for focus $(0, p)$ and directrix $y = -p$ is

\sqrt{x^2 + (y - p)^2} = \lvert y + p \rvert.

Squaring cancels $y^2$ and $p^2$ and leaves $x^2 = 4py$ . The number $p$ tells you where the focus is.

Example For $x^2 = 12y$ we have $4p = 12$ , so $p = 3$ . The focus is $(0, 3)$ and the directrix is $y = -3$ .

Shifted equations

Definition Moving the vertex to $(h, k)$ gives

(x - h)^2 = 4p(y - k), \qquad (y - k)^2 = 4p(x - h).

The first has focus $(h, k + p)$ and directrix $y = k - p$ . The second has focus $(h + p, k)$ and directrix $x = h - p$ . Any equation $y^2 + Dy + Ex + F = 0$ with $E \ne 0$ , or $x^2 + Dx + Ey + F = 0$ with $E \ne 0$ , can be put in one of these forms by completing the square.

In plain words Replace $x$ by $x - h$ and $y$ by $y - k$ , and the whole picture slides over by $(h, k)$ .

Example Take $y^2 - 4y - 8x - 4 = 0$ . Then $y^2 - 4y + 4 = 8x + 8$ , which is

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

The vertex is $(-1, 2)$ and $p = 2$ , so the focus is $(1, 2)$ and the directrix is $x = -3$ .

Focal property

Definition For a point $P = (x, y)$ on $x^2 = 4py$ with $p > 0$ , the focal distance is

\lvert PF \rvert = y + p,

because $\lvert PF \rvert$ equals the distance from $P$ to the directrix $y = -p$ , and $y \ge 0$ . The chord through the focus perpendicular to the axis, the latus rectum, has endpoints $(\pm 2p, p)$ and length $4\lvert p \rvert$ .

In plain words You can read off the distance to the focus without a square root, just by looking at the height of the point.

Example For $x^2 = 8y$ we have $p = 2$ . The point $(4, 2)$ is on it, and its distance to the focus $(0, 2)$ is $y + p = 4$ , which matches the direct calculation $\sqrt{16 + 0} = 4$ . The latus rectum runs from $(-4, 2)$ to $(4, 2)$ and has length $8$ .

Reflection property

Definition Let $P$ be a point on a parabola other than the vertex, with focus $F$ . The tangent line at $P$ makes equal angles with the segment $PF$ and with the line through $P$ parallel to the axis. Consequently, a ray traveling parallel to the axis that reflects off the inside of the parabola passes through $F$ , and a ray from $F$ reflects into a ray parallel to the axis.

In plain words Light coming in straight along the axis always bounces to the focus. That is why dishes and headlights have this shape.

Example Take $x^2 = 4y$ with focus $F = (0, 1)$ and the point $P = (2, 1)$ . The slope there is $\frac{x}{2} = 1$ , so the tangent is $y = x - 1$ . It crosses the axis at $T = (0, -1)$ . Now $\lvert FP \rvert = 2$ and $\lvert FT \rvert = 2$ , so triangle $FPT$ is isosceles, and its base angles at $P$ and $T$ are equal. The angle at $T$ equals the angle between the tangent and the vertical line through $P$ , since that line is parallel to the axis. So the two angles at $P$ match.

Geometric construction

Definition Given the focus $F$ and the directrix $L$ , choose any point $Q$ on $L$ . The line through $Q$ perpendicular to $L$ meets the perpendicular bisector of the segment $FQ$ at exactly one point $P$ , and $P$ lies on the parabola. Moreover, that perpendicular bisector is the tangent line to the parabola at $P$ . Letting $Q$ run along $L$ produces every point of the parabola.

In plain words Points on the perpendicular bisector of $FQ$ are equally far from $F$ and $Q$ . The point $P$ above $Q$ is then equally far from $F$ and from the line, which is exactly the defining property.

Example Take $F = (0, 1)$ , $L$ the line $y = -1$ and $Q = (4, -1)$ . The midpoint of $FQ$ is $(2, 0)$ and $FQ$ has direction $(4, -2)$ , so the bisector is $4(x - 2) - 2y = 0$ , or $y = 2x - 4$ . On the vertical line $x = 4$ this gives $P = (4, 4)$ . Indeed $4^2 = 4 \cdot 4$ , so $P$ lies on $x^2 = 4y$ , and the slope there is $\frac{x}{2} = 2$ , the slope of the bisector.

Check yourself

Find the focus and directrix of $y^2 = -8x$ .

Show the answer

Comparing with $y^2 = 4px$ gives $4p = -8$ , so $p = -2$ . The focus is $(-2, 0)$ and the directrix is $x = 2$ . The parabola opens to the left.

Write the equation of the parabola with vertex $(1, 3)$ and focus $(1, 5)$ .

Show the answer

The axis is vertical and $p = 2$ , so $(x - 1)^2 = 8(y - 3)$ .

On $x^2 = 12y$ , how far is the point $(6, 3)$ from the focus?

Show the answer

Here $p = 3$ , so the distance is $y + p = 6$ .

Parabolas

Key terms

Focus and directrix

Vertex and axis

Standard equations

Shifted equations

Focal property

Reflection property

Geometric construction

Check yourself

How this lesson connects

Builds on

Used later in