PARABOLA: AUXILIARY ELEMENTS
In the previous lesson, we studied parabola
and its graphs for different cases. Today
we will analyze parabola's auxiliary elements:
Directrix, Focus, Vertex. We will also learn
how to construct the locus of vertices L(x)
of a moving parabola P(x) = a x^2 + b x +
c, when a and c fixed while b is varying.
(1) CONSTRUCTING A PARABOLA
A parabola is the set of points that are equidistant from a fixed line and a fixed point not on the line. The line is called the DIRECTRIX of the parabola, and the point is called the FOCUS. In the previous lesson, we learned that the extremum point of a parabola is called its VERTEX.
Here is an outline of constructing a parabola in GSP:
- Create two points.
- After selecting those points, use the Construct Menu to create a line that passes through those points. (Remember from basic geometry that you need at least two distinct points to draw a line). This line will be the DIRECTRIX.
- Select the line and open the Construct Menu, and choose Point on Object. This will create a point on the directrix which will be animated later.
- Now select both the line and the point. Construct a perpendicular line using the Construct Menu.
- Click the point tool to draw a point that is NOT on the directrix. This point will be the FOCUS.
- Now select both the focus and the moving point on the directrix. Use Construct Menu to draw a line segment.
- Select the segment and create its midpoint.
- Select the segment AND the midpoint. Use Construct Menu to draw a perpendicular line. This line represents the perpendicular bisector of the line segment that connects the focus and the moving point on the directrix.
- Now select both the perpendicular bisector AND the perpendicular line that contains the moving point on the directrix. Then use Construct Menu or simply right click: This will create the intersection point. In GSP literature, this point is called a trace point.
- Now the most important part: Select the trace point and right click: Enable the Trace Point option. This is that point that will trace the parabola.
- Finally select the moving point and right click. Choose the Animate Point option.
- Click the picture below.
As you see, when we animate the moving point,
the trace point traces a parabola. Observe
that the distance between Focus and the Trace Point is always the same as the distance between
the Trace Point and the Moving Point: This is by definition.
(2) LOCUS OF VERTICES OF A MOVING PARABOLA: GENERAL CASE
The second activity today, will be the generation of the locus of vertices L(x) of a moving parabola P(x) = a x^2 + b x + c, when a and c fixed while b is varying.
In the previous lesson, we learned that any
parabola of the form y = P(x) = a x^2 + b
x + c can be written as y = P(x) = a ( x
- r )^2 + k , where r = - b / 2a , and ,
k = c - b^2 / 4a. Let's rewrite this as (
y - k ) = a ( x - r )^2: We translated the
parabola to the r - k axis. The point ( r
, k ) , which is the extremum point of the parabola, is also called VERTEX.
The focus point of the parabola ( y - k ) = a ( x - r )^2 has coordinates ( r , k + 1/4a ). The equation of its directrix is y = k - 1/4a.
Observe that r = - b / 2a is a function of a and b only, while, k = c - b^2 / 4a is function of a, b, and c. This means that if we let a parabola P(x) = a x^2 + b x + c move when a and c fixed while b is varying, then its vertex moves as well. What will be the trajectory of the vertex? This is the key question we'll study in this section.
To answer this question, we have to find a way to relate the coordinates of the vertex, namely r and k: Use their definiton: r = -b / 2a and k = c - b^2 / 4a. I have to find a way to express k as function of r:
r = -b / 2a => b= - 2ar => b^2 = 4 a^2 r^2 => k = c - a r^2
This is another parabola's equation! Another parabola that does not contain b. This is another parabola that depends ONLY on the constants a and c, nothing else. Why?
In the above equation k = c - a r^2, k and r are dummy variables. Therefore we are free to use any letter we want. Let's rewrite it as y = c - a x^2. This curve is called the LOCUS OF VERTICES of the original parabola P(x) = a x^2 + b x + c . Let's give it a name: L(x) = c - a x^2. We are done! The following little simple if-then reasoning summarizes this section:
If P(x) = a x^2 + b x + c , then L(x) = c - a x^2
(3) LOCUS OF VERTICES OF A MOVING PARABOLA: SPECIAL CASE OR APPLICATION
As an application, let us fix a = 1 and c = 1 , and vary b between -6 and +6 . Now simply refer to the if-then reasoning we deduced above:
If P(x) = 1x^2 + b x + 1, then L(x) = 1- 1x^2
We are done! L(x) = 1 - x^2. Click the Animate Parameter button to see the trajectory of L(x): It's a parabola.
(4) PREPARE FOR NEXT LESSON
Next week we will study the centers of a triangle in detail. For the moment, just for an application, let's draw a triangle in GSP and locate its centroid, circumcenter, orthocenter, and incenter. When we fix one side of the triangle (say |AB| ) and let the opposite corner (C) move on a line that is parallel to the fixed side, we get an animation that includes these centers with all possible triangles (right, obtuse, acute, etc.) In the animation, let us make sure that we enabled trace point option for the centers. The paths of the circumcenter and the centroid are not very interesting: straight lines. But the incenter follows an ellipse and the orthocenter follows a parabola.
Download GSP file here.