Perpendicular tangents

Calculus Level 3

(α1,β1), (α2,β2),  , (αn,βn)(\alpha_1, \beta_1), \ (\alpha_2, \beta_2), \ \cdots \ , \ (\alpha_n, \beta_n) are the points on the curve S : y=sin(x+y)  x[4π, 4π]S \ : \ y=\sin (x+y) \ \ \forall x \in [-4\pi, \ 4\pi]
such that the tangents at these points to SS are perpendicular to the line l : (21)x+y=0.l \ : \ (\sqrt {2}-1)x+y=0.

Find the value of k=1nαkβk.\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.

Details and Assumptions

  • \lfloor{\cdots}\rfloor denotes the floor function (greatest integer function).

  • Even though in the problem, a plural form of points is given, there may only be one such point that exists.


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