Parabolas

The distance condition that defines a parabola

Fix a point $F$ and a line $d$ that does not pass through $F$. The parabola with focus $F$ and directrix $d$ is the set of all points $P$ in the plane satisfying

that is, every point whose distance to $F$ equals its distance to $d$. The point $F$ is the focus, and the line $d$ is the directrix. The whole curve is a locus: a set of points picked out by a condition, not by a formula.

The condition that $F$ is not on $d$ matters. If you put the focus on the directrix, the only points that stay equidistant are the ones on the perpendicular to $d$ through $F$. In that case the “curve” collapses into a straight line. A parabola needs the focus held off the line.

Two more names come straight out of the definition. The axis of the parabola is the line through $F$ perpendicular to $d$. Reflecting the plane across that axis swaps nothing, because it fixes $F$ and fixes $d$, so it maps the curve onto itself. The vertex is the point of the curve on that axis. The vertex must be the midpoint between $F$ and the directrix, since that is the one point on the axis whose two distances agree.

To turn that picture into algebra you need to measure two distances, so here are the two measuring tools, both of which come from the Pythagorean theorem.

Distance between two points. Put $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ and complete the right triangle whose legs are horizontal and vertical. The horizontal leg has length $|x_2 - x_1|$, the vertical leg has length $|y_2 - y_1|$, and $PQ$ is the hypotenuse, so the Pythagorean theorem gives

You have already met this formula in disguise: the modulus $|a + bi| = \sqrt{a^2 + b^2}$ is exactly the distance from $a + bi$ to the origin in the complex plane.

Distance from a point to a line. The distance from a point to a line means the shortest distance, and it is measured along the perpendicular. Any other segment from $P$ to the line is the hypotenuse of a right triangle whose leg is that perpendicular. A hypotenuse always beats a leg, so no other route can be shorter. For a horizontal line $y = c$ the perpendicular is vertical, so the distance from $P = (x_0, y_0)$ is simply

and for a vertical line $x = c$ it is $|x_0 - c|$. Nothing more complicated is needed, because every directrix in this lesson is horizontal or vertical.

Deriving the equation from the definition

Now put the definition to work in coordinates. Place the vertex at the origin, and let $p$ be the signed distance from the vertex up to the focus, so that

The vertex $(0,0)$ is then correctly halfway between them. Everything about the curve is now encoded in the single number $p$.

Read the result again, because it is the punchline of the lesson. Dividing by $4p$ turns the equation into

which is $y = ax^2$ with $a = \dfrac{1}{4p}$. The purely geometric locus, defined without a single coefficient, is the quadratic graph you have been drawing since Chapter 4. Run the relation the other way and every quadratic $y = ax^2$ acquires a focus and a directrix it always had but never advertised:

For $y = x^2$, for instance, $a = 1$ gives $p = \tfrac14$: that familiar curve is the set of points equidistant from the point $\left(0, \tfrac14\right)$ and the line $y = -\tfrac14$.

The sign of $p$ carries the direction. If $p > 0$ the focus sits above the vertex, the directrix below, and the curve opens upward, wrapping around its focus. If $p < 0$ the focus sits below the vertex, the directrix above, and the curve opens downward. In both cases the parabola bends toward the focus and runs away from the directrix, which is the fastest way to get the direction right without memorizing anything.

The definition also hands you a shortcut worth keeping. For a point $(x,y)$ on $x^2 = 4py$ with $p > 0$, its distance to the focus equals its distance to the directrix by definition, and the second of those is easy:

No square root, no substitution. The distance from a point on the parabola to the focus, called its focal distance, is read straight off the $y$-coordinate.

The focal width, and why the number is $4p$

Where does the $4$ in $x^2 = 4py$ actually show up on the picture? Draw the chord through the focus parallel to the directrix and measure it. On $x^2 = 4py$ the focus is at height $y = p$, so substitute:

The chord runs from $(-2p,\, p)$ to $(2p,\, p)$, so its length is $|4p|$. That chord is the focal width (older books call it the latus rectum), and it is the single most useful number for sketching a parabola by hand. Go to the focus, move $|2p|$ each way, and you have two more points on the curve.

This also settles the most common confusion in this lesson. The coefficient in $x^2 = 4py$ is $4p$, and it is not the focus distance. It is the focal width. The focus distance is a quarter of it. A curve with a big $|4p|$ is wide and shallow, and its focus sits far from the vertex. A curve with a small $|4p|$ is narrow and steep, and its focus is tucked in close.

Moving the vertex

Nothing so far depended on the vertex being at the origin, and moving it costs no new work. Translation is a rigid motion: sliding two points by the same vector does not change the distance between them. In the same way, sliding a point and a line by the same vector does not change the distance from the point to the line. So if you translate the focus and the directrix by $(h, k)$, the locus of equidistant points translates by $(h,k)$ as well. That follows because the defining condition is stated entirely in terms of distances.

Apply that to the focus $(0,p)$ and directrix $y = -p$, whose locus is $x^2 = 4py$. Translating by $(h,k)$ sends the focus to $(h, k+p)$, the directrix to $y = k - p$, and every point $(x,y)$ of the locus to $(x + h, y + k)$. Writing the new coordinates as $X = x + h$ and $Y = y + k$, so that $x = X - h$ and $y = Y - k$, the equation $x^2 = 4py$ becomes the standard form of a parabola with a vertical axis:

This is the bridge back to Chapter 4. Any quadratic $y = ax^2 + bx + c$ can be put into vertex form $y = a(x-h)^2 + k$ by completing the square, and rearranging that gives $(x-h)^2 = \tfrac{1}{a}(y - k)$, which is the standard form above with $4p = \tfrac{1}{a}$. Every parabola in the focus-directrix sense with a vertical axis is the graph of a quadratic, and the graph of every quadratic is a parabola in the focus-directrix sense. The two definitions describe the same curves, which is exactly why one name is used for both.

Notice the shape of that conversion: the square must stand alone on one side. As soon as you write $y = a(x-h)^2 + k$ you are one step away, but the step matters, because $4p = \tfrac1a$ and not $a$.

Parabolas that open sideways

Nothing in the definition says the directrix has to be horizontal. Take a vertical directrix instead, $x = -p$, with the focus at $(p, 0)$. The derivation is the same computation with the roles of $x$ and $y$ exchanged: a point $(x,y)$ is on the curve when

and squaring, expanding, and cancelling the $x^2$ and $p^2$ terms leaves

Shifting the vertex to $(h,k)$ by the same translation argument gives the standard form of a parabola with a horizontal axis:

Now $p > 0$ opens the curve to the right and $p < 0$ opens it to the left, and the focal width is still $|4p|$.

Say plainly what this curve is not. $y^2 = 4px$ is not the graph of a function of $x$. Solving for $y$ gives $y = \pm 2\sqrt{px}$, which is two different values at every $x$ past the vertex and none at all on the other side. So a vertical line meets the curve twice or misses it entirely, and the vertical line test fails. It is still a perfectly good parabola under the geometric definition, which never mentioned functions; it simply cannot be written as $y = f(x)$. Turn your head sideways and it is a function of $y$, namely $x = \tfrac{1}{4p}\,y^2$.

Why the focus deserves its name

A parabola does something no other shape does, and it is the reason the focus is called the focus. Every ray that arrives parallel to the axis leaves the curve through that one point. Every ray. A radio dish collects a plane wave from a satellite millions of kilometres away and concentrates it onto a receiver the size of a thumb. A telescope mirror gathers starlight and delivers it to the eyepiece. A car headlight runs the same argument backwards, putting the bulb at the focus so the reflected light leaves in a parallel beam.

The proof needs one fact from physics, that a ray reflects off a curve at equal angles to the tangent line. The proof also needs one fact from geometry, that the set of points equidistant from two points is the perpendicular bisector of the segment joining them. That second fact is worth pausing on, because it is the same kind of locus as the parabola’s own definition. That locus is built from a point and a point, instead of a point and a line.

This is also why the shape cannot be faked in practice. A shallow circular mirror is nearly parabolic near its centre. So such a mirror will bring the rays that land near the middle roughly to a point, but the further out a ray strikes, the further it misses. What the proof gives you is that a parabola sends every parallel ray through the focus, however far out it strikes. That property is exactly what a telescope mirror or a dish antenna needs, which is why they are ground and pressed to that curve. The converse is true as well, that the parabola is the only curve with this property, but nothing above proves it.

Common mistakes