Although we define the parabola as an intersection of a cone and plane parallel to the edge of the cone, this is pretty clearly a difficult definition to work with.
A parabola is the locus of all points equally distant between a point and line.
But luckily it turns out that there is a proof (due to Dandeline, which can be found in a nice presentation here: Dandeline's proof) that a parabola is equivalently defined as the locus of points equally distant between some point and some line.
More precisely, we begin with a point which we call the focus, and a line which we call the directrix. In principle these may be any point and line which are not coincident.
For simplicity we will assume (until indicated otherwise) that the focus has horizontal coordinate equal to 0. This will make calculations simpler. The only thing left unknown about the focus, then, is its vertical coordinate.
We will write the focus as .
Also for simplicity we assume that the directrix is horizontal. Therefore the directrix always has the form
where
We also define the vertex to be the point which is equally distant from the focus to the line. This makes the vertex either the minimum point (if the directrix is below the focus) or the maximum point (if the directrix is above).
Exercise 1. Find the vertex.
Suppose the focus is and the directrix is . Find the vertex.
Solution
A parabola is the locus of all points equally distant between a point and line.
The diagram to the right shows essentially this situation, although coordinates are not shown explicitly.
Still, if the focus is and the vertex is visibly straight down from this by a distance of 2, then the vertex is .
Exercise 2. Tilt your head and find the vertex
We will primarily consider parabolas for which the directrix is horizontal. In this case we often say that the resulting parabola "opens vertically".
However, once you understand parabolas which open vertically, it is not very hard to understand ones which open horizontally (and have a directrix which is vertical).
Suppose the focus is and the directrix is . Find the vertex.
Solution
Focus at (0,0), directrix at x=1.
As with most things in algebra and geometry, your first instinct should always be to draw a picture to help understand the problem. I've drawn the picture to the right.
As always with parabolas, the vertex is just precisely half-way between the focus and directrix.
In this case, it is V(1/2, 0).
For a given focus and directrix, they determine the parabola with points equally distant from focus and from directrix.
In general, a point is on the parabola if and only if the distance FP equals the distance from P to the directrix.
Because F(0,c) then the distance FP is given by the distance formula,
Because the directrix is horizontal, it is easy to find the distance to P.
By definition of a parabola, being points at which these two distances are equal, then
We define p to be the distance from the vertex to the focus and therefore, trivially, it is always half the distance from the focus to the directrix. It can always be computed by
If we write the coordinates of the vertex as then clearly because the focus, too, has its horizontal coordinate at 0. Also, if we assume then
If on the other hand then .
Exercise 3. Derive the Parabola Formula
From the setup described above, derive the equation
.
Hint: Write out the equation using what you know about each of these. Then square both sides to get rid of the square-root. Then distribute out all operations until you're able to solve for .
Solution
and
so setting these equal, we can cancel the on each side,
Solving for we get
which, by the difference of squares, is
Recalling that and then we get
The right-hand side is . If we factor the then we're done.
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Exercise 4. Now a full pass through
Suppose the focus is at and the directrix is .
Find all other information. The full set of information in any parabola problem is: the values of , the coordinates of the vertex, the equation of the directrix, and the equation of the parabola.
Solution
Since then we are given c=2. Since the directrix is then we are given . Since there is no horizontal translation .
With these values known we can state the locations of all points.
The directrix was given as .
Finally, the equation of the parabola is which simplifies to
Exercise 5. Horizontal translation
Let's consider what happens when the focus does not have its horizontal coordinate at 0. (It will turn out: not much.)
Suppose the focus is at and the directrix at . Find the equation of the parabola determined by these.
Solution
We find many of the relevant quantities the same way as before. We have and so . Also . The equation would usually be which is
This problem is no different except that, because the focus is translated 2 to the left, then the relation is translated 2 to the left.
We've seen earlier that to translate the relation by , it is represented by . Here we translate by .
Therefore the equation of this translation is , which is
Until now we've assumed the directrix is horizontal, and the focus is above the directrix. This causes the parabola to open upward.
If the focus is below the directrix, all the same analysis applies. You will find that the only thing which changes is
which is necessary so that p is positive. This is necessary because p is a distance and so must be positive.
Then downstream of that, we get an equation which is only slightly different.
Exercise 7. Downward Parabola
Suppose the vertex of a parabola is at and the focus is at . Determine the equation of the parabola.
Solution
It's given that and , and . Then gives
so . Also gives
Then the equation of the parabola is which is
Next consider if the directrix is vertical rather than horizontal. Now the vertex and focus will share the same vertical coordinate by may have different horizontal coordinates. We now say that and are the coordinates of the focus and vertex, and as before p is the distance from focus to vertex.
Much like before,
If the focus is to the right of the directrix, the equations become
and the equation becomes
If the focus is left of the directrix,
and the equation becomes
A consolidated display of the four cases is shown below.
vertical, horizontal / focus greater, lesser
Opens
Focus greater
Focus lesser
vertical
horizontal
Exercise 8. Parabolic dish
A satellite dish
A satellite dish is in the shape of a parabola, as shown.
It turns out that, if a light ray comes into the dish and strikes the surface, it will always reflect off of the surface and be directed into the focus. This is shown on the left.
Assume that the rim of the dish has a diameter of 20 feet. Also assume that the depth of the dish is 10 feet.
Find the point at where all of the incoming rays will intersect.
The red rays represent light rays coming into the dish, reflecting off of the surface, and then being directed into the focal point.
Solution
In effect we need to find the distance from the the vertex to the focus. The focus is the point where the rays intersect, and the vertex is the "bottom" of the depth of the parabola.
It shouldn't matter where we place the satellite in our coordinate space, so just for simplicity, let's set the vertex at (0,0) and have the parabola open upward. So . Then the equation is
Because of this choice of coordinate system, the right edge of the rim is 10 feet to the right on the x axis, and 10 feet above the axis. That is to say, it passes through the point .