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Problem 82 Hard Difficulty

A tangent line is drawn to the hyperbola $ xy = c $ at a point $ P. $
(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is $ P. $
(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where $ P $ is located on the hyperbola.

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Video Transcript

Yeah, it's clear someone named Reed here. So we have. Why is equal to see next to the negative one power. We can rewrite it as follows so the derivative, when we find it, you get negative C x to the negative to power the tangent line Therapy of X one comma See over X one is why minus c over X one over X minus X one is equal to negative C next one the negative to power and this becomes why is equal to negative c x of one the negative to power X plus two c ex of one, the negative one power. So the Y intercept iss to see over at someone that iss the Y coordinates of P Um, that's two times the white cornet and you got the X intercept. We make zero equal negative C except one too negative to power. Thanks plus two C texts of one to the negative one power, and that gives us to accept one. And that's twice the X coordinates of P. So P is the midpoint off the line segment formed between the intercept for part B. We got that Why is equal to see X to the negative one. So the derivative is equal to negative. C x, the negative to power we want The tangent line simplifies to why is equal to negatives CX of 12 negative to Power X plus two c ex someone the negative one power. So the why intercept is to see over X someone when we get the X intercept by making it zero and we've bought two x someone we know that the area of the triangle is equal to have of the X intercept times the Y intercept, which is to see divided by ex someone on two ex of one is equal to to see and that's independent of X sub one so p.