The Parabola as a locus of points
A parabola can be defined as the locus of a points such that its distance from a fixed point (called the focus) is equal to its distance from a fixed line (called the directrix).
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- Click on the Directric and Point button to construct a point on the directrix. Click and drag the point - notice that the line does not change direction or location but the point will move along this directrix.
- Click on Construct Focus and move it where you want it.
- Click the Measure Distances, Show Trace Point and Construct Point of Locus buttons. Notice that the distance from the focus to the point of the locus is equal to the distance from the locus point to the point on the directrix.
- Now drag the point on the directrix back and forth and watch the pattern that is formed. It appears to be a parabola. Clear the screen using the red X in the bottom right of the JavaWindow. Now move the focus point further or closer from the directrix and drag the point on the directrix again. Click the red X again to clear the traced lines.
- Press the Show Animate Button-button, then the Hide Buttons. Animate the sketch and watch the sketch trace out a locus of points and create a parabola.
- Stop the animation by clicking on the Animate button again. Clear the screen using the red X, click Show buttons, Show Traceline, Hide Buttons and Animate again!
- Try to formulate a proof that this figure is indeed a parabola.
Definitions:
- A locus of points is a collection of points that satisfies a particular requirement.
- A directrix is a fixed line that serves as a guide in creating our parabola
- A focus is the point used to determine the parabola's openness and distance from the directrix.
- A parabola is a collection of points (a locus) such that the moving locus point is always equidistant from the focus and the directrix.