Solution assignment 08 Tangent line to graph

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Assignment 8

The line:

y=x

intersects the parabola:

y=x^2-7x+12

in two points A en B.
The tangent lines at these points intersect in a point C.
Calculate the coordinates of C.

Solution

First we calculate the intersection points, thus we have to solve the following equation:

x^2-7x+12=x

We rewrite this:

x^2-8x+12=0

and we can solve this equation by factorizing:

x^2-8x+12=(x-6)(x-2)=0

and thus:

x=2 or x=6

The corresponding intersection points are:

(2,2) and (6,6)

In order to determine the slopes of the tangent line(s) at these points we have to differentiate the function of the parabola:

y'=2x-7

For the slope of the tangent line at (2,2) we find m=-3 and for the one at (6,6) we find m=5. The equation of one tangent line is:

y-2=-3(x-2)

or:

y=-3x+8

The equation of the other tangent line is:

y-6=5(x-6)

or:

y=5x-24

We find the intersection point of these tangent lines by solving the equation:

-3x+8=5x-24

The solution of the equation is x=4 and the corresponding y-value is y=-3\cdot4+8=-4. The point C has the coordinates (4,-4).

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