Tangential Network 1

Calculus Level 5

Let \(n\) be a real number; define the curve \(C_n\) by the equation

\[x^n + y^n = 1.\ \ \ \ x, y \geq 0\]

For every point \(x, y\) on \(C_n\) there is a tangent line. This tangent line intersects the coordinate axes in points \((\xi,0)\) and \((0,\eta)\), respectively.

There exists one curve \(C_n\) for which the value of \(\xi + \eta\) is the same for all these tangent lines. How much is \(n\)?


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