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 December 19th, 2009, 11:58 PM #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 : 2x-3y=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, 12:31 AM #2 Global Moderator   Joined: Dec 2006 Posts: 19,047 Thanks: 1618 No. Your guess was incorrrect. There are two bisectors.
December 20th, 2009, 04:43 AM   #3
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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:
 gradient of the bisector line : (-3/4 x 2/3) x 1/2 = -1/24
Should be gradient of the bisector line : (-3/4 + 2/3) x 1/2 = -1/24

December 20th, 2009, 07:23 AM   #4
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Re:

Quote:
 Originally Posted by skipjack No. Your guess was incorrrect. There are two bisectors.
Thanks both of you , but how do i find the other one ? Are the 2 bisectors perpendicular to each other ?

 December 20th, 2009, 01:58 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, 04:23 PM #6 Global Moderator   Joined: Dec 2006 Posts: 19,047 Thanks: 1618 What are the equations of the angle bisectors of the lines with equations 3x + y = 6 and x - 3y = 2?
 December 20th, 2009, 05:29 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, 12:17 AM #8 Global Moderator   Joined: Dec 2006 Posts: 19,047 Thanks: 1618 Correct. Hence mikeportnoy can get correct results using that method.
December 21st, 2009, 02:07 AM   #9
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Re: Two straight lines

Hi mikeportnoy;

Quote:
 Originally Posted by skipjack Correct. Hence mikeportnoy can get correct results using that method.
Following skipjack's problem forced me to do some research on this problem. The page that I originally found:

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:

$y=(18 - 5 \sqrt{13}) (x - 2)\text{ \ and \ } y= (18 + 5 \sqrt{13}) (x - 2)$

The formula for the bisectors is found and proved here:

http://www.tutorvista.com/content/ma...s/bisector.php

 December 21st, 2009, 07:53 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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