Consider the hyperbola $xy=1$ with $x>0$ and $y>0.$ The tangent line at any point on the hyperbola forms a right triangle with the coordinate axes. Show that the area of the triangle does not depend on the point chosen. To show this, do the following:
a) Choose a point on the hyperbola of your choice. Find the tangent line and then find the area of the right triangle described above.
b) Choose a different point of your choice. Find the tangent line and then find the area of the right triangle described above.
c) Now for the general case, choose the point at $x=a.$ Find the tangent line at $x=a$ and then find the area of the right triangle described above.
This is the work I have so far:
No method to find the area of a right triangle was given, so I just used 1/2*B*H.
But I get stuck when given $x=a$ for part C.
Any ideas?
Best Answer
Since this is apparently an assignment for credit, I won’t say any more than I would if it were my assignment, and you came to my office. You need to check your work for (a) and (b): you’ve found the slopes of the tangent lines correctly, but your lines don’t pass through the points $\langle 1,1\rangle$ and $\langle 2,0.5\rangle$, so they aren’t the actual tangent lines. As a result, you’ve not found the $x$- and $y$-intercepts correctly, and your base and height numbers are wrong. Once you fix those, you should do exactly the same thing for (c), but using the point on the hyperbola whose $x$-coordinate is $a$. The first step is write down the $y$-coordinate of that point.