Mutual Tangent Lines
This problem illustrates a method you can use to find lines that are tangent to two curves.
Consider the quadratic functions
f(x) = x2+ 10
and
g(x) = −(x−8)2+ 9
If we sketch their graphs on the same axes, we can see that there are two lines that are tangent to
both curves:
-100
-50
0
50
100
-25 -20 -15 -10 -5 0 5 10 15 20 25
Let’s find the equations of those lines. Consider either line.
Suppose it is tangent to y=f(x)at the point (a, b). Suppose it is tangent to y=g(x)at the point
(c, d).
This looks like we are introducing four variables: a, b, c, and d. But in fact there are only really
two: since (a, b)is on the graph of y=f(x), we have
b=f(a)
Similarly,
d=g(c)
So we just need to find the values of aand c.
If we could write two equations involving aand c, we’d be in business (we’d have ”two equations
in two unknowns”, and that’s a good place to be algebraically).
The way to get the two equations is to write the slope of the tangent line in several ways.
