Exploring Analyic Geometry with Mathematica®

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Radical Axis of Two Circles

radaxis.html

Exploration

Show that the two circles "radaxis_1.gif" and "radaxis_2.gif" have the radical axis x-y=0.

Approach

Convert the equations to circles and find the radical axis of the circles.

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 the circles from the equations.

Clear[a,b,c];
{C1,C2}={Circle2D[Quadratic2D[1,0,1,a,b,c]],
         Circle2D[Quadratic2D[1,0,1,b,a,c]]}

"radaxis_3.gif"

Construct the radical axis.

L1=Line2D[C1,C2] //Simplify

"radaxis_4.gif"

Convert the line to an equation.

Clear[x,y];
Equation2D[L1,{x,y}]

"radaxis_5.gif"

Discussion

This is a plot of a numerical example with a=1, b=5 and c=-1.

Sketch2D[{C1,C2,L1} /. {
   a->1, b->5, c->-1}]

"radaxis_6.gif"

Graphics saved as "radaxi01.eps".


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