October 4th, 2012, 01:33 AM  #1 
Newbie Joined: Oct 2012 Posts: 9 Thanks: 0  Analytic Geometry Circle
Show that the circles x² + y²  4x + 2y  20 =0 and 2x² + 2y²  17x + 16y + 65 = 0 touch internally and find the point of contact. Another similar question : Prove that the circles x² + y² + 2x 8y + 8 = 0 and x² + y² +10x  2y + 22 = 0 touch one another externally and find the point of contact. Please help. I do not understand what it means by the circles touching internally and externally. How to solve both this questions. What method should I use? Thanks 
October 4th, 2012, 09:23 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Analytic Geometry Circle
1.) Here is a plot of the two circles, which shows what is meant by touching internally: [attachment=0:ag4o32ow]internaltouch.jpg[/attachment:ag4o32ow] To find the point where they touch, we may observe that this point must lie on the line passing through the centers of the two circles, so we may write the circles in the form: We see that this circle is centered at We see that this circle is centered at The slope of the line passing through the two centers is Using the pointslope formula, we find the line passing through the two centers is: Now, using the equation of the first circle, we may substitute for : We know the must be on the interval so we must take the positive root: And so: And so the point at which the two circles touch is . 
October 4th, 2012, 08:27 PM  #3 
Newbie Joined: Oct 2012 Posts: 9 Thanks: 0  Re: Analytic Geometry Circle
Thank You for the clear explanation. For the circles to touch externally, all we have to do is find the gradient that cuts through the diameter of the two circles, is that right?

October 4th, 2012, 08:29 PM  #4 
Newbie Joined: Oct 2012 Posts: 9 Thanks: 0  Re: Analytic Geometry Circle
The way to solve both equations is the same? It doesn't matter whether it is internally or externally?

October 4th, 2012, 08:52 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Analytic Geometry Circle
Yes, the method is the same. I neglected to show that the circles touch internally, which may be done by showing the distance between the centers of the circles is equal to the larger radius less the smaller one. For the second problem, you can show they touch externally by showing the distance between the centers is equal to the sum of the radii. 
October 4th, 2012, 10:48 PM  #6 
Newbie Joined: Oct 2012 Posts: 9 Thanks: 0  Re: Analytic Geometry Circle
Thanks Mark!


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