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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

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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   May 3, 2020
edited by CPhill  May 3, 2020