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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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).

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