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.
+128475 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
