**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 =>

This is another parabola's equation! Another parabola that does not contain b. This is another parabola that depends

In the above equation k = c - a r^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.