My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 4th, 2012, 02:33 AM   #1
Newbie
 
Joined: Oct 2012

Posts: 9
Thanks: 0

Analytic Geometry- Circle

Show that the circles x + y - 4x + 2y - 20 =0 and 2x + 2y - 17x + 16y + 65 = 0 touch internally and find the point of contact.

Another similar question : Prove that the circles x + y + 2x -8y + 8 = 0 and x + y +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
Maths4ever is offline  
 
October 4th, 2012, 10:23 AM   #2
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 522

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 point-slope 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 .
Attached Images
File Type: jpg internaltouch.jpg (22.3 KB, 277 views)
MarkFL is offline  
October 4th, 2012, 09:27 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?
Maths4ever is offline  
October 4th, 2012, 09:29 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?
Maths4ever is offline  
October 4th, 2012, 09:52 PM   #5
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 522

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.
MarkFL is offline  
October 4th, 2012, 11:48 PM   #6
Newbie
 
Joined: Oct 2012

Posts: 9
Thanks: 0

Re: Analytic Geometry- Circle

Thanks Mark!
Maths4ever is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
analytic, circle, geometry



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
analytic geometry condemath Geometry 1 August 29th, 2011 04:50 AM
Analytic Geometry Q3 Rayne Geometry 0 November 2nd, 2010 07:32 AM
Analytic Geometry Q2 Rayne Geometry 0 November 2nd, 2010 06:57 AM
Analytic geometry help please nortpron Geometry 6 April 5th, 2010 02:07 AM
Analytic Geometry amkroes Geometry 12 November 7th, 2009 04:53 PM





Copyright © 2019 My Math Forum. All rights reserved.