Tangential Network 2

Calculus Level 5

Let nn be a real number; define the curve CnC_n by the equation xn+yn=1.    x,y0x^n + y^n = 1.\ \ \ \ x, y \geq 0 For every point x,yx, y on CnC_n there is a tangent line. This tangent line intersects both coordinate axes. Consider the line segment between these intersection points.

There exists one curve CnC_n for which all these tangential line segments have the same length. How much is nn?

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