How to prove that hyperbola y=4/x determines a triangle which area is 8 when I draw tangent point at whatever point on this hyperbola and when I consider that x-axis, y-axis and tangent line form triangle.
Let be on . For simplicity sake assume,Originally Posted by totalnewbie
.
Now the tangent line to point is the derivative at that point. Thus, is the slope of the derivative. Now find the equation of the tangent.
The equation of line having slope and point is (by point slope formula is):
Now the x-intercept is when . Thus,
Thus,
Thus,
Thus, it intercepts the x-axis at .
This would imply that the base of the triangle must be
And its height to be . Thus, the area must be a constant for whatever value of .
Similarily it is true when . For one thing this hyperbola is symettric.
Q.E.D.
Hello,Originally Posted by ThePerfectHacker
I'm a little bit puzzled: You wrote: This would imply that the base of the triangle must be
And its height to be . Thus, the area must be a constant for whatever value of .
1. As I understand the problem the base of the right triangle must be 2x'.
2. You get the interception with the y-axis if x=0. Then the height of the triangle is .
If you take these results the value of the area will be 8 as totalnewbie has written.
Greetings
EB
Okay let me start again the triangle is formed by the intersecting line between the y-axis the x-axis.
Since the equation of the of tangent at point in my other post is,
.
Now, the x-axis and y-axis are formed when y is zero and x is zero respectively.
Thus,
When y=0
Then,
Thus,
When x=0
Then,
Thus,
Now, the area is,
Q.E.D.
Here is diagram below.