Question

How do you show that the graphs of the two equations y=x and y=1/x have tangent lines that are perpendicular to each other at their point of intersection?

Answer 1

Please see below.

Explanation:

The graphs of `y=x` and `y=1/x` intersect at `x=1/x` or

`x^2=1` or `x=+-1`

Now as `y=x` is a straight line, its slope for `y=x` is `1` everywhere.

Slope of `y=1/x` will be given by its derivative i.e. `-1/x^2`.

Hence at both the points given by `x=+-1`, the slope of `y=1/x` is `-1`.

Hence at both points given by `x=+-1`, the product of slopes of two equations is `-1` and hence

Graphs of the two equations `y=x` and `y=1/x` have tangent lines that are perpendicular to each other at their point of intersection.

graph{(y-x)(y-1/x)=0 [-10, 10, -5, 5]}