Show that every triangle formed by the coordinate axes and a tangent line to y = 1/x ( for x > 0) always has an area of 2 square units.

Hint: Find the equation of the tangent line at x = a. (It will contain a’s as well as x and y.) Then find the

x-and y-intercepts for that line to find the lengths of sides of the right triangle.

Step-by-step explanation:

given a point the equation of a line with slope m that passes through the  given point is

or equivalently

.

Recall that a line of the form , the y intercept is b and the x intercept is .

So, in our case, the y intercept is and the x  intercept is .

In our case, we know that the line is tangent to the graph of 1/x. So consider a point over the graph . Which means that

The slope of the tangent line is given by the derivative of the function evaluated at . Using the properties of derivatives, we get

. So evaluated at we get

Replacing the values in our previous findings we get that the y intercept is

The x intercept is

The triangle in consideration has height and base . So the area is

So regardless of the point we take on the graph, the area of the triangle is always 2.