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Two perpendicular lines l1 and l2 in the xy-plane intersect at the point (7, 5). If the sum of the slopes of l1 and l2 is 10, find the sum of their y-intercepts.

 May 3, 2020
 #1
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Let  the slope of the first line  =  m

And let the slope of the second line  =  -1/m

 

And  the sum of their slopes  = 10   which implies that

 

m - 1/m  = 10       multiply through by m

 

m^2 - 1  = 10m

 

m^2  - 10m - 1  =  0

 

m^2 - 10m  =  1

 

m^2 - 10m + 25  =  1 + 25

 

(m - 5)^2  =  26

 

m - 5  =  √26

 

m =  √26 + 5

 

-1/m  =   -1/ [ √26 + 5]  =  - [√26 - 5]   =  5 - √26 

 

For the first line we  have that

 

y = (√26 + 5) ( x -7) + 5 

 

When x  = 0   the y intercept  is

 

y = (√26 + 5) (-7)  + 5     =   - 30 - 7√26

 

 

And for the second line we have that 

 

y =  (5 - √ 26) ( x - 7) + 5

 

When x  = 0   the y intercept   is

 

y = (5 - √26) ( -7) + 5  =   -30 + 7√26

 

Here's  a graph to show this  : https://www.desmos.com/calculator/g3ayap2tgm

 

And the sum of the y intercepts  =  -60

 

cool cool cool

 May 3, 2020
edited by CPhill  May 3, 2020

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