In this activity, we'll look at a geometric approach to constructing a parabola. Specifically, we'll investigate how a parabola can be described as a [b]locus of points[/b].
In geometry, a [i]locus of points [/i]is a set of all the points that satisfy one or more specified conditions. For example, a circle is the locus of points that are the same distance from a fixed point (the centre). In the interactive diagram below, drag the red point [b][color=#cc0000]A[/color][/b] to see how a circle of radius 4 cm is the locus of points that have a distance of 4 cm from the point [b]C[/b].
Consider the following interactive diagram.[br]The position of the blue point [b][color=#1e84cc]F[/color][/b] is fixed (doesn't move). It's called the [b][color=#1e84cc]focus[/color][/b].[br]The blue line [b][color=#1e84cc]d[/color][/b] is also fixed. It's called the [b][color=#1e84cc]directrix[/color][/b].
Let's look for a relationship based on the lengths of the two dashed line segments above ([b]AF[/b] and [b]AB[/b]). Specifically, let's investigate the locus of points such that the two dashed line segments have the same length. To do so, follow these steps:[br][br][list=1][*]Drag the red point [b][color=#cc0000]A[/color][/b] to any position such that the two dashed line segments appear to have the same length.[/*][*]Left-click the red point [b][color=#cc0000]A[/color][/b] to record its position on the diagram.[/*][*]Repeat steps 1 and 2 for several other possible locations for the red point [b][color=#cc0000]A[/color][/b] (at least 20). Be sure to include points on both the left and right sides of the focus (point [b][color=#1e84cc]F[/color][/b]).[/*][/list]
What type of curve do your recorded points appear to outline?
What is the significance of the point directly between the focus and the directrix?
It's the vertex of the parabola.
[i][b]Note: [/b]The interactive diagram below may be helpful in answering some of the following questions. You can drag the focus and directrix to reposition them. Drag the background to pan and use the mouse wheel to zoom. [/i][list=1][*]How would moving the focus further above the directrix affect the vertex of the parabola?[/*][*]How would moving the focus below the directrix affect the shape of the parabola?[/*][*]How does increasing the space between the focus and the directrix affect the shape of the parabola?[/*][*]How could you set up the focus and directrix to create a parabola that opens to the left? What about one that opens to the right?[/*][*]How could you set up the focus and directrix to create a [i]slanted parabola[/i]?[br][/*][/list]