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December 20th, 2009, 12:58 AM   #1
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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??
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December 20th, 2009, 01:31 AM   #2
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No. Your guess was incorrrect. There are two bisectors.
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December 20th, 2009, 05:43 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
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December 20th, 2009, 08:23 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 ?
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December 20th, 2009, 02:58 PM   #5
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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
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December 20th, 2009, 05:23 PM   #6
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What are the equations of the angle bisectors of the lines with equations 3x + y = 6 and x - 3y = 2?
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December 20th, 2009, 06:29 PM   #7
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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
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December 21st, 2009, 01:17 AM   #8
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Correct. Hence mikeportnoy can get correct results using that method.
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December 21st, 2009, 03:07 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:

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
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December 21st, 2009, 08:53 AM   #10
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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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