Tangential Curve

Calculus Level 5

Let t[0,1]t \in [0, 1] be a parameter. Let t\ell_t be the line passing through the points (t,0)(t,0) and (0,1t)(0,1-t).

For each value of aa we can define the point PtP_t where t\ell_t intersects the neighboring lines t+dt\ell_{t+dt}.

The collection of points PtP_t traces out a curve, as suggested in the drawing above. This curve satisfies the equation xn+yn=1x^n + y^n = 1. Determine the value of nn.

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