find equations of 2 tangent lines passing through (2,5)

f(x)= 4x-x^2

points of tangent line are (3,3) and (1,3)

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- February 5th 2010, 05:23 PMphalangeequations of the 2 tangent lines
find equations of 2 tangent lines passing through (2,5)

f(x)= 4x-x^2

points of tangent line are (3,3) and (1,3) - February 5th 2010, 05:49 PMArchie Meade
- February 6th 2010, 04:04 AMHallsofIvy
Since you are given the points of tangency, that is particularly easy. If you were only given that the tangent lines went through (2, 5), it would be harder. Suppose the point of tangency is . The derivative at is 4- 4x_0 so the equation of the tangent line there is . At x= 2, that is or . That gives x= 1 and x= 3 as the points of tangency.

Here is Fermat's pre-calculus (literally!) method:

Any non-vertical line through (2, 5) can be written as y= m(x-2)+ 5. In order that such a line be tangent to , they must, of course, touch: [tex]y= 4x- x^2= m(x-2)+ 5[tex] for some x= a. That is the same as saying has x=a as solution and so has (x-a) as a factor.

To be**tangent**a must be a**double**root and x-a must be a**double**root. That means that we must have

for all x.

Set coefficients equal and solve for m.

m- 4= -2a and . Since m= 4- 2a, 2m= 8- 4a and or as before. - February 6th 2010, 11:57 AMArchie Meade
That was beautifully portrayed, HallsofIvy!

thank you,

much appreciated!