Question

For some k>0 the lines 50x+my=1240 and my=8x+544 intersect at right angles at the point (m,n).find m+n.

Answer 1

the slope of the first line is =50/m
the slope of the 2nd line is 8/m
if they are perpendicular , the product of their slope is -1,
or the slopes are negative reciprocals of each other
that is 50/m = -m/8
so m^2 = -400, which is not possible

or

since (m,n) lies on both lines,
50m + mn = 1240
mn = 8m + 544

substitution:
50m + 8m + 544 = 1240
58m = 696
m = 12
12n = 96 + 544
n = 53 1/3 or 160/3

however, that would make the first equation:
50x + 12y = 1240 with a slope of -50/12 or -25/6
and the second equation:
12y = 8x + 544 with a slope of 8/12

the lines would NOT be perpendicular

You have a contradiction in your question.