Exploring Analyic Geometry with Mathematica®

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Medial Curve Type



Show that the medial curve equidistant from a point and a circle is a hyperbola when the point is outside the circle, and it is an ellipse when the point is inside the circle.  (Hint: Examine the value of the discriminant "mdtype_1.gif" of the medial quadratic.)


Create the expression "mdtype_2.gif" from the coefficients of the medial quadratic. Consider "mdtype_3.gif" with the circle at the origin. Show that the expression is negative when the point is inside the circle and positive when the point is outside the circle.


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.



Set the coefficients of the quadratic (from equations listed in the book).


Find the discriminant, "mdtype_4.gif", at the origin.

disc=b^2-4*a*c /. {h2->0,k2->0} //Simplify


Using the distance formula, the point is outside the circle then the discriminant is positive, implying a hyperbola; if the point is inside the circle than the discriminant is negative, implying an ellipse.

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