PARABOLA : Exploration

Properties

When the parabola is studied as the graph of a function, it is pictured in the ‘upright’ position as at right. This is useful for studying its properties in terms of the function y = f(x) = x2, or more generally in terms of y = f(x) = ax2 + bx + c. However, this is not our purpose here, and we shall consider it placed in the ‘horizontal’ position below.

We associate with the parabola the point F (the focus) on the x-axis, and the line d (the directrix) perpendicular to the x-axis. The focus and the intercept of the directrix with the x-axis are equidistant from the origin, and the actual common distance is determined by the parabola. The x-axis here is called the axis of the parabola – the line of symmetry, and O is the vertex of the parabola.

Let us now investigate some interesting properties of the parabola.



Focus-directrix property

Click the adjacent figure to open the applet which shows a parabola with its focus and directrix. Click the ‘Animate’ button and watch what happens. Now stop the animation (click the button again), and try manually dragging the point on the y-axis.

State the property that you see illustrated here. If your statement involves the word ‘circle’, rephrase your statement so that this word does not occur.

We have illustrated here the basic property of the parabola: a parabola is the locus of points which are equidistant from a fixed point (focus) and a fixed line (directrix).







Reflection property

Now look at this next applet. Think of there being a light source at the focus F, and the light reflecting off the parabola. Run the animation and then manually move the point on the y-axis.

What do you notice about the reflected rays? How do these lines relate to the tangent line at the point on the parabola? (You might like to use your protractor!) State a result about the parabola which is demonstrated here.

This property of the parabola is the reason torches and lamps and satellite dishes have parabolic reflectors. Any light emanating from the focus, is reflected by the parabola in a direction parallel to the axis. If we draw in the tangent t at any point Q of the parabola, the angle made with t by the line QF to the focus F, is equal to the angle made by t to the line through Q parallel to the axis (‘angle of incidence = angle of reflection’).



The normal property

Click the figure at right and run the applet, using both the ‘Animate’ button and manually.

Describe the two red segments through the point Q on the parabola. Now look carefully at the segment NG. What do you notice about this segment? Can you make a conjecture about the length of NG? In terms of any other length in the diagram?

This is an interesting result. Through Q we have drawn the perpendicular to the tangent line meeting the x-axis in G, and also the normal from Q to the x-axis, meeting the axis in point N. Surprisingly, the length of NG is constant, and is in fact equal to twice the length of OF.

This gives a good method of constructing a tangent to the parabola.





Parallel chord property

This property concerns any set of parallel chords of a parabola. Click the figure at right to open the applet which has two distinct parts. There is the main parabola figure, and then at lower left there is a small circle. The point D can be chosen anywhere on the circle, determining the segment CD, the slope of which gives the slope of the parallel chords. You can use the 'Animate' button, or adjust the action manually using the red point on the y-axis.

Look at the parallel chords Qq. Point M denotes the midpoint of the chord. You will notice that this point has a very nice trajectory. What is it? Look at what happens for different positions of D. For example, what is the path of M when CD is parallel to the y-axis?

We see that for any set of parallel chords, the midpoints of the chords lie on a line parallel to the axis of the parabola.

This applet was quite challenging to write. You might like to investigate the line through F which is normal to the line Qq. It passes through an interesting point. What is it?



The three tangents property

Click the figure at right to open the applet. This applet illustrates two interesting properties of the triangle formed by three tangents to the parabola.
First click the ‘Animate’ button to juts get a feel of the triangle. Points P, Q, R move arbitrarily on the parabola. You will notice three corresponding red points on the y-axis: these can be used to manually manipulate the position of the points P, Q, R. The triangle we are interested in is UVW.

Click the ‘Show circumcircle ’ button. Check that it is in fact the circumcircle of UVW. What else do you notice about this circle? When you have finished, hide the circumcircle.

Now click the ‘Show orthocentre’ button. What is the orthocentre? What rather amazing property is illustrated here?

We observe two surprising results here. Firstly, the circumcentre of the triangle of tangents always passes through the focus F of the parabola. Secondly, the orthocentre of this triangle always lies on the directrix of the parabola.

This makes one wonder what other surprising results lie in store for one who explores the parabola!