Given the function f(x) = (2x)/(x-4), determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through A(5,10) and B(8,4).

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llltkl | College Teacher | (Level 3) Valedictorian

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Given `f(x) = (2x)/(x-4)`

`f'(x)=((x-4)(2x)'-2x(x-4)')/(x-4)^2`

`=(2(x-4)-2x*1)/(x-4)^2`

`=(2x-8-2x)/(x-4)^2`

`=-8/(x-4)^2`

Let the coordinates of the required point on f(x) be (h,k). 

Slope of the tangent line at a point (h,k) is f'(h).

`f'(h)=-8/(h-4)^2`

Slope of the secant line that passes through the points A(5,10) and B(8,4) is:

`=(4-10)/(8-5)`

`=-6/3`

`=-2`

By the condition of the problem,

`-8/(h-4)^2=-2`

`rArr (h-4)^2=(-8)/(-2)=4`

`rArr h-4=+-2`

`therefore h=6, 2`

When `h=6, k=(2*6)/(6-4)=12/2=6`

When `h=2, k=(2*2)/(2-4)=4/(-2)=-2`

So, the required point has coordinates of (6, 6) or (2, -2).

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