Medial Curve Type

mdtype.html

Exploration

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 of the medial quadratic.)

Approach

Create the expression from the coefficients of the medial quadratic. Consider 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.

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

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

Clear[x1,y1,h2,k2,r2];
a=4*((x1-h2)^2-r2^2);
b=8*(x1-h2)(y1-k2);
c=4*((y1-k2)^2-r2^2);

Find the discriminant, , 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.