# Proof With Area Problem

• May 2nd 2008, 03:30 AM
finch41
Proof With Area Problem
enjoy:

consider the function y = 1/x
let A be a point on the function
the tangent through A intersects the coordinate axes at C(c,0) and D(0,d)
prove that the area of this triangle is the same no matter where A is located and find that area
• May 2nd 2008, 03:37 AM
Moo
Hello,

Did you draw a sketch ?

The area of the triangle OCD is defined by $\frac{OD \cdot OC}{2}$. Do you agree ?
Distance OD is d and distance OC is c, so $\mathbb{A}=\frac{cd}{2}$

Now, you may know the equation of the tangent to a function. Let m be the absciss of A.
The equation will be :

$y=f'(m)(x-m)+f(m)$

Knowing that $f'(x)=\frac{-1}{x^2}$, determine the equation.

You know that C and D are on this tangent. Hence, replace their coordinates in it and isolate c and d.

So you will have $c=\dots$ and $d=\dots$ with respect to m.

Substitute in the expression of $\mathbb{A}$ and conclude :)

It's long because I've given all the elements, so try to do and when you can't move on, post :D
• May 2nd 2008, 04:15 AM
finch41
its not that hard is it

why oh why didnt i write this on my test?

i managed to do it in 5 minutes on the subway ride home

• May 2nd 2008, 04:16 AM
Moo
Quote:

Originally Posted by finch41
its not that hard is it

why oh why didnt i write this on my test?

i managed to do it in 5 minutes on the subway ride home