   Chapter 2.3, Problem 110E

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# Sketch the parabolas y = x 2 and y = x 2 − 2 x + 2 Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

To determine

To sketch:

The parabolas y = x2 and y = x2 - 2x + 2 and check if there is a line that is tangent to both curves.

Explanation

Calculation:

Consider the curve  y=x2,

By differentiating, the slope  of tangent at x is dydx=2x.

The second curve is y=x2-2x+2 , which gives the slope of tangent at x as dydx=2x-2.

Let y=mx+b  be the equation of line tangent to y=x2 at point (x1 , x12 )

and tangent to y=x2-2x+2 at  x2 ,  x22-2 x2+2 .

This gives us following equations

m=2x1 =2x2-2

x12=m x1+b  and

x22-2x2+2=m x2+b

From the first equation we’ve x1=m2  and x2=m+22= m2+1

Plug x1=m2  in the second equation  x12=m x1

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