Finding points where tangent lines touch curve

Question

Hello! If someone could help me with this problem I would greatly appreciate it!

1) There are two tangent lines to the curve y = x/x+2 passing through the point (1,2).Find the x

-coordinates of the points where these tangent lines touch the curve. Use exact answers. Enter answers as a comma separated list.

The x-coordinates are ________

Answer 1

Every point on the graph of y will be of the form (x, x/(x+2)). The slope of a line passing through a point with those coordinates and the point (1,2) will look like this:

m = ( 2 - x/(x+2) ) / (1-x)

The derivative of y (dy/dx) is y' = 2/(x+2)². That is also a formula for the slope of a line with an x coordinate of x.

So these two expressions must be equal:

(2 - x/(x+2) ) / ( 1 - x) = 2/(x+2)².

Here you probably want to cross multiply, ending with a quadratic equation which you can solve for x using the quadratic formula--giving the required to values for x.

Try the above and check it out here:

desmos.com/calculator/n8utwh9qbm