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Answer 1 · 2018-05-02T17:04:59

#c = 61/4# and #c= -65/4#.

Normal means perpendicular to the curve. The given normal line can be rewritten as follows:

#y = 1/2x + c/2#

This means that the slope is #1/2#. The slope of the tangent at that point will be #-1/(1/2) = -2#.

Thus we must find the values of #x# where the derivative equals #-2#.

#y = 4(2x - 1)^-1#

The derivative as given by the chain rule is

#y' = -8/(2x- 1)^2#

Set #y' = -2#.

#-2 = -8(2x- 1)^2#

#1/4 = (2x- 1)^2#

#+- 1/2 = 2x - 1#

#2x= 3/2 and 2x = 1/2#

#x = 3/4 and x = 1/4#

The curve's corresponding y-values are given by calculating using the function

#y(3/4) = 4/(2(3/4) - 1) = 4/(1/2) = 8#
#y(1/4) = 4/(2(1/4) - 1) = 4/(-1/2) = -8#

We now solve for #C# knowing that #2y= x + c#.

#2(8) = 3/4 + c -> c= 16 - 3/4 -> c = 61/4#
#2(-8) = 1/4 + c-> c = -16 - 1/4 -> c = -65/4#

Hopefully this helps!