Expert: Paul Klarreich - 9/20/2007

Question
Let (a,b) be an arbitrary point on the graph y=1/x, x>0. Prove that the area of the triangle formed by the tangent line through (a,b) and the coordinate axes is 2.

Answer
Questioner:   Austin
Category:  Calculus
Private:  No
 
Subject:  Calculus-Tangent Lines, Area
Question:  Let (a,b) be an arbitrary point on the graph y=1/x, x>0. Prove that the area of the triangle formed by the tangent line through (a,b) and the coordinate axes is 2.
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Hi, Austin,

Note:  your arbitrary point is  (a,1/a), since y = 1/x

First part: Determine the equation of this tangent line.

dy/dx = -x^-2 = -1/x^2

At  (a,b),  m = -1/a^2

Use the point-slope form:

y - y0 = m(x - x0),  where  (x0,y0) is  (a,1/a)

y - 1/a = -1/a^2(x - a)

Simplify a bit:

y - 1/a = -x/a^2 + 1/a
y  = -x/a^2 + 2/a

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Second part: Determine the x- and y-axis intercepts.

Set  x = 0:

y = 2/a

Set y = 0:

0 = -x/a^2 + 2/a

x/a^2 = 2/a

x = 2a^2/a = 2a

Now your triangle has  base = 2a, the x-intercept, and height = 2/a, the y-intercept.

Area = 1/2 base * height = 1/2(2a)(2/a) = 2