The Focal Property of a Parabola - Maple Programming Help

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The Focal Property of a Parabola

Main Concept

In mathematics, a locus is a set of points whose location satisfies specified conditions.

If we plot a point P, called the focus, and a line l, called the directrix, that does not pass through P, the locus of points equidistant from P and l forms a parabola. In other words, it is the graph of a quadratic equation.

Explanation

Suppose the point P  has Cartesian coordinates xP , yP and l is a horizontal line with the equation y = yl . We can do this without loss of generality by simply rotating the coordinate system appropriately. Now choose any point in the plane with the general coordinates x, y.

The distance from P to x, y is given by dP=xxP2+yyP2.

The distance from l to x, y is given by dl=xx2+yyl2=yyl2.

So, equating these distances and solving for y, we find:

yyl2 = xxP2+yyP2 

yyl2 = xxP2+yyP2  

 y2  2yly + yl2  =  x2  2 xPx + xP2 + y2  2 yPy + yP2   

 2yly  + 2 yPy  =  x2 2 xPx + xP2  + yP2  yl2 

y =  x2 2 xPx + xP2  + yP2  yl22yl + 2 yP

Since all of the indexed variables are constants, this is simply a quadratic equation, proving that the locus of points equidistant from P and l forms a parabola.

 

Click and drag the red point in the plot below to set the focus, P, and use the slider to change the position of the directrix l. Then, choose the "Find Equidistant Points" radio button and click on the plot to add some points which are equidistant from P and l. Use the "Show Parabola" check box to plot the parabola and see its equation. Click "Reset" to reset the graph.

 

 

Distance from P to Latest Point =

Distance from l to Latest Point =

Equation of Parabola:

Position of Line

 

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