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December 20th, 2009, 12:58 AM  #1 
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  Two straight lines
Given two straight lines L_1 : 3x+4y=6 and L_2 : 2x3y=4. Find the equations of the bisectors of the angles between L_1 and L_2. My working: gradient of L1 : 3/4 gradient of L2 : 2/3 gradient of the bisector line : (3/4 x 2/3) x 1/2 = 1/24 Then the intersection of L1 and L2 : (2,0) So the equation would be y=(1/24)x+1/12 Am I correct?? 
December 20th, 2009, 01:31 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,154 Thanks: 1421 
No. Your guess was incorrrect. There are two bisectors.

December 20th, 2009, 05:43 AM  #3  
Senior Member Joined: Dec 2009 From: Las Vegas Posts: 209 Thanks: 0  Re: Two straight lines
Hi mikeportnoy; You have correctly found one bisector, as skipjack points out there is another one. Also you have a small typo; Quote:
 
December 20th, 2009, 08:23 AM  #4  
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  Re: Quote:
 
December 20th, 2009, 02:58 PM  #5 
Newbie Joined: Dec 2009 Posts: 1 Thanks: 0  Re: Two straight lines
Indeed, the two lines are perpendicular to one another. You may use the identity m1m2 = 1 for perpendicular lines to find the gradient of the second bisector and then use the point of linersection of L_1 and L_2 to calculate the equation of the second line Hope this helps, mintsmike 
December 20th, 2009, 05:23 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,154 Thanks: 1421 
What are the equations of the angle bisectors of the lines with equations 3x + y = 6 and x  3y = 2?

December 20th, 2009, 06:29 PM  #7 
Senior Member Joined: Dec 2009 From: Las Vegas Posts: 209 Thanks: 0  Re: Two straight lines
Hi skipjack; I used the method on this page to do your tricky problem: http://www.tutorvista.com/content/ma...s/bisector.php I have y =  ( x / 2 ) + 1 and y = 2 x  4 
December 21st, 2009, 01:17 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,154 Thanks: 1421 
Correct. Hence mikeportnoy can get correct results using that method.

December 21st, 2009, 03:07 AM  #9  
Senior Member Joined: Dec 2009 From: Las Vegas Posts: 209 Thanks: 0  Re: Two straight lines
Hi mikeportnoy; Quote:
http://answers.yahoo.com/question/in...8054307AAOxUju Which stated that the bisectors were found by an arithmetic mean of the slopes of the 2 lines was apparently wrong. The correct bisectors are: The formula for the bisectors is found and proved here: http://www.tutorvista.com/content/ma...s/bisector.php  
December 21st, 2009, 08:53 AM  #10 
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  Re: Two straight lines
So my way of finding the equation of bisector by taking the mean of the slopes is wrong? Thanks Bobby, I will go through them. 

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