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There wasn't a mistake, just one more step was needed: Is there a method to do this division, and how do you get the intuition to divide it anyway? o_O

Oh I see! Using the answers from WolframAlpha gives me the right results. Now to find that mistake! Thank you all!

Oh yes, the ##x_{1}## is supposed to be squared. I tried averaging them but it became very cumbersome

I am simultaneously solving 1) The equation of the hyperbola ##y-y_{1}=\frac{b^{2} x_{1}}{a^{2} y_{1}}\left(x-x_{1}\right)## with the equation of the top asymptote ## bx + ay = 0## 2) The equation of the hyperbola ##y-y_{1}=\frac{b^{2} x_{1}}{a^{2} y_{1}}\left(x-x_{1}\right)## with the...

Summary:: Question: Show that the segment of a tangent to a hyperbola which lies between the asymptotes is bisected at the point of tangency. From what I understand of the solution, I should be getting two values of x for the intersection that should be equivalent but with different signs...

Yikes! YES, thanks for the catch! I was so frustrated trying to solve this one.

I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T? Answer: I just can seem to get to this. I think I'm there but can't get it in...