# Thread: Proof With Area Problem

1. ## 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

2. Hello,

Did you draw a sketch ?

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

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

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

Knowing that $\displaystyle f'(x)=\frac{-1}{x^2}$, 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 $\displaystyle c=\dots$ and $\displaystyle d=\dots$ with respect to m.

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

It's long because I've given all the elements, so try to do and when you can't move on, post

3. 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

4. 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

What do you mean ? lol
It's not really hard, no... You just have to know the definitions..

I hope this won't penalize you too much...