Making Sense of the Directrix and

Focus of a Parabola!

How many times have you been given the definitions of terms like the locus, the directrix, and the focus of a parabola? Did you memorize the definitions so that you might regurgitate them on a test? Did you ever fully understand what they were? If you think that you understood it when you were 17, do you still understand?

The definitions seem quite simple:

parabola: The locus of points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are equal.

So, from this definition, you now know exactly what a parabola is.

Oh, wait. What's a locus you ask? It is merely a set of points satisfying a condition. From the definition above, it is the parabola.

I want you to be able to "see" this so I am going to call on my old friend, Geometer's Sketchpad. First, I want to show the x-axis and the y-axis, so I highlight "show grid." This is not necessary, but maybe more helpful at placing your directrix and focus. Place an arbitrary point on your graph, the focus, and an arbitrary line, the directrix. I chose my directrix to be the x-axis and my focus is the point (3,2). Place a "movable" point on the directrix, construct a segment from that point to the focus, and construct the midpoint of that new segment. This ensures that this point on the directrix is equidistant from the focus. Now, when we move the point along the directrix, the midpoint of this segment will always keep the focus equidistant from that particular point on the directrix; however, the distance from a point to a line is found by drawing a perpendicular line through the point to the line. Now, it is necessary to draw this perpendicular line through our movable point to the directrix (which is my x-axis). The point where they intersect, is that magical point where the focus and directrix are equidistant. When we plot a set of these points, we have a beautiful parabola!

To see this as an animation, click here.

Now, for those of you that have had Calculus, you may notice a special relationship above in the blue line and the parabola. The blue line is the tangent line to the parabola. This allows us to make even more connections involving the first derivative of a function (we are specifically looking at a quadratic function).

To see this as an animation, click here.

There are fun ways to develop other conic sections like the ellipse and the hyperbola using Geometer's Sketchpad.

To see this as an animation, click here.

To see this as an animation, click here.

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