Exploring Analyic Geometry with Mathematica®

Home Contents Commands Packages Explorations Reference
Tour Lines Circles Conics Analysis Tangents

Circle Tangent to Two Lines, Radius

tancir3.html

Exploration

Show that the centers (h,k) of the four circles tangent to the perpendicular lines

"tancir3_1.gif" and "tancir3_2.gif"

with radius r=1 are given by

"tancir3_3.gif",

"tancir3_4.gif",

"tancir3_5.gif",

"tancir3_6.gif".

Assume that the two lines are normalized, "tancir3_7.gif".

Approach

A circle "tancir3_8.gif" tangent to a line A x+B y+C=0 implies that

"tancir3_9.gif"

giving two equations.  The fixed radius r=1 is a third equation. Solve three equations in three unknowns.

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

Solve the three equations.

Clear[r,h,k,A1,B1];
ans1=Solve[{r^2==( A1*h+B1*k)^2,
            r^2==(-B1*h+A1*k)^2,
            r==1},
           {h,k,r}]

"tancir3_10.gif"

Simplify.

ans2=ans1 //. A1^2+B1^2->1

"tancir3_11.gif"


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