Question 992326
the two circles are:
(x-3)^2 + (y-4)^2 = 25
(x-1)^2 + (y-5/2)^2 = 225/4


if they touch, then the tangent line to each circle will pass through the point that they touch at.


the radius of each circle will be perpendicular to the tangent line at that point.


this means the two radii will form a straight line that goes through the point of tangency and will intersect both circles at the point of tangency.


the line will be a straight line formed by the line segment between the two centers of the cirfcle.


the centers of each circle are (3,4) and (1,(5/2)


using these two points, you get a straight line with the equation of y = 3/4 * x + 7/4.


that line should intereset with both circles at the same point.


i solved it graphically to see that the point of intersection is (7,7).


that graph is shown below:


<img src = "http://theo.x10hosting.com/2015/092603.jpg" alt="$$$" </>


it should be able to be solved algebraically but i didn't do it because i ran out of time and it's messy.


i did verify algebraically that the point (7,7) satisfies both equations, so there's no question that is the point of intersection of the two circles.



you would probably want to solve the intersection of that line with each equation using the substitution method.


graphing was much easier.