Find all points (x, y) on the graph of y=x/(x-7) with tangent lines perpendicular to the line y=7x-2

The slope of a tangent to a curve y at any point is given
by the value of the derivative of the curve at that
point.

We have y = x/(x -
7)

y'= -7/(x - 7)^2

The line y
= 7x - 2 has a slope of 7. All line perpendicular to this have a slope of
-1/7.

y' = -7/(x - 7)^2 =
-1/7

=> (x - 7)^2 =
49

=> x - 7 = 7 and x - 7 =
-7

=> x = 14 and x =
0

y = 2 and y =
0

The points on the curve where the slope is
perpendicular to y = 7x - 2 are (14, 2) and
(0,0).