
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 26th, 2010, 02:03 AM  #1  
Newbie Joined: Jul 2010 Posts: 1 Thanks: 0  Tangents for a Circle
I have found this exercise on the internet, but I just can't figure it out. I tried it with vectors Quote:
But how should I continue? Thanks in advance!!  
July 26th, 2010, 02:55 AM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Tangents for a Circle
My first instinct would be to draw the circle, the point, and the tangent lines. Use Pythagorean theorem to get the length of the segments (ie sqrt20). You should be able to see that this problem is identical to finding the intersection of two circles: A centered at the origin, radius 2, standard equation x^2 + y^2 = 4 B centered at (0,4), radius sqrt20, standard equation ______ Solve this system of equations and you're done! 
July 26th, 2010, 03:50 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,622 Thanks: 2074 
No, you're not, as the question asked for the equations of the tangents, not their points of contact. Also, your calculated radius was incorrect. Each equation must have the form y = mx + 4. Substituting for y in the equation x² + y² = 4 gives x² + (mx + 4)² = 4, i.e., (m² + 1)x² + 8mx + 12 = 0. This equation's discriminant must be 0, so 16m²  12(m² + 1) = 0, i.e., m² = 3. Hence the tangents' equations are y = ?3x + 4 and y = ?3x + 4. 
July 26th, 2010, 05:23 AM  #4  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Tangents for a Circle Hello, much12! Did you make a sketch? Quote:
Code: A o (0,4) /\ /  \ /  \ /  \ /  \ / * * * \ /*  *\ C o  o B * *  * * *  * 2 * *  * *   *     *O    *   [color=beige]. . [/color] [color=beige]. . [/color]  
July 26th, 2010, 06:12 AM  #5 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Tangents for a Circle
Guess I should have drawn the picture... SQRT(12)! And while the form "must" be y = mx + 4, the general case requires using the distance between the centers to find the radius of the second, and then solving a system of equations. (algebraically) 
July 26th, 2010, 08:10 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,622 Thanks: 2074 
There's nothing particularly special about this case, where it wasn't necessary to find the distance you refer to.

July 26th, 2010, 08:48 AM  #7 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Tangents for a Circle
I thought the centers both being on the yaxis was kinda special!

July 26th, 2010, 01:38 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,622 Thanks: 2074 
That makes your initial distance calculation simpler, but has no significant effect on the method I used.


Tags 
circle, tangents 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Tangents to 2 circles  Jimi  Algebra  3  November 27th, 2013 04:25 AM 
Common tangents to circle and parabola  Daltohn  Algebra  8  October 23rd, 2013 02:37 AM 
2 Tangents and 1 parabola  suuup  Calculus  2  October 31st, 2012 06:46 PM 
Can someone help me with secants and tangents?  gebraroest  Calculus  3  April 18th, 2011 02:03 PM 
Equations of All Tangents  Calc12  Calculus  9  December 14th, 2010 12:15 PM 