Exploring Analyic Geometry with Mathematica®

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Tangent to an Ellipse with Slope

ellslp.html

Exploration

Show that the lines tangent to the ellipse "ellslp_1.gif" with slope m are given by "ellslp_2.gif".

Approach

Construct a line with slope m and use the function TangentLines2D[ln,quad] to construct the desired tangent lines.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct a line with slope m.

Clear[x,y,m];
l1=Line2D[Point2D[x,y],m]

"ellslp_3.gif"

Construct the lines tangent to the ellipse and parallel to the line.

Clear[a,b];
tln=TangentLines2D[l1,e1=Ellipse2D[{0,0},a,b,0]]

"ellslp_4.gif"

Show the lines in equation form.

Map[Equation2D[#,{x,y}]&,tln]

"ellslp_5.gif"

Discussion

Plot a numerical example.

Sketch2D[{tln,e1} /.
         {m->1/2,a->2,b->1}]

"ellslp_6.gif"

Graphics saved as "ellslp01.eps".


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
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