SOLUTION: There are two tangent line to the curve y=4x-x^2 that pass through the point (2,5). find the equations of these two lines and make a sketch to verify your results

Question 982609: There are two tangent line to the curve y=4x-x^2 that pass through the point (2,5). find the equations of these two lines and make a sketch to verify your results
Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
Slope of the tangency line must be same as dy/dx for the parabola, and this dy/dx will be a variable expression.
This slope based on the parabola:

Again, this is a SLOPE for the as yet unknown tangent lines.

Here is a look just at the parabola:

You should be able to understand that (2,5) is on the axis of symmetry and is 1 unit above the vertex.

The tangent line built using Point-Slope equation form:
This must contain the given point (2,5);

and doing the algebra steps....
----Understand this without its appearance fooling you. This represents a LINE, but it is variable.

If you sketch the graph, the understanding should intuitively be this:
We must have (tangent line) - (parabola) = 0, in order for the intersection to be only any single point.
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Use the formulas of y for each of those objects.

Do the algebra steps...
omitting them here...
----------This will give the x coordinates of the two tangent lines at the points of tangency on the parabola.
... and it is FACTORABLE.

x=1 or x=3

Use each individually in the line equation , which is equation for line intersection the parabola. ONLY substitute for the slope, which is ; so that you can get the equation of the TANGENT line.
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Doing that and simplifying :

and

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The two tangent lines.

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