This applet demonstrates the focus-directrix definition of a parabola, that is:[br][br]A [b]parabola[/b] is the locus of a point [i]P[/i] in a plane such that the distance between [i]P[/i] and the focus [i]F[/i] is equal to the distance between [i]P[/i] and the directrix.[br][br]In other words, if [i]M[/i] is the foot of the perpendicular from [i]P[/i] to the directrix, then[br][center][i]PF[/i] = [i]PM[/i].[/center]

[u]How to use the applet:[br][/u]Click on the buttons in the top row to see the following parabolas:[br][list][*][b][i]y[/i][sup]2[/sup] = 4[i]ax[/i][/b] - Focus ([i]a[/i],0), directrix [i]x[/i] = -[i]a[/i][/*][*][b][i]x[/i][sup]2[/sup] = 4[i]ay[/i][/b] - Focus (0,[i]a[/i]), directrix [i]y[/i] = -[i]a[/i] [/*][/list][br]Click on [b]Reset view[/b] to return the value of [i]a[/i] to 1.[br][br]Click and drag on the [b]slider[/b] to change the value of [b]a[/b].[br][br]Click and drag on the [b]point [/b][i]M[/i] to see how the point [i]P[/i] moves to satisfy the condition [i]PF[/i] = [i]PM[/i].[br]Click on [b]Animate[/b] to move the point [i]M[/i] automatically. Click on [b]Stop[/b] to stop the animation.[br][br]Click on [b]Trace On[/b] to trace out the locus of the point [i]P[/i] as you move the point [i]M[/i]. Click on [b]Trace Off[/b] to stop tracing. To remove the trace from the screen, click and drag the window slightly.[br][br]After observing the locus of [i]P[/i], click on the check box [b]Show parabola[/b] to see the parabola for the given focus and directrix.