Hello, prasum!
Here's a back-door approach . . .
Two circles each of radius 5 units touch each other at (1,2).
If the equation of common tangent is ,
find the equation of two circles. Code:
*
|\ * * *
| \ * *
-4 | \ * *
| \ * *
*- - *
3 \ * P *
\ * ♠ *
\* o | *
* * * \ o |
* ** o | *
* * - - - ◊ - - - * *
* | o ** *
| o \ * * *
* | o *\
* ♣ * \
* Q * \
\
* * \
* * \
* *
* * *
The two circles are tangent at: .
The centers of the circles are: .
The tangent has slope
The radii, are perpendicular to the tangent
. . and have length 5.
From move 4 units right and 3 units up to
. . and we find that: .
From move 4 units left and 3 units down to
. . and we find that: .
Hence, the centers of the circles are: .
Therefore, their equations are: .