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For which values of n does the line x/a + y/b = 2 act as a tangent to the curve (x/a)ⁿ + (y/b)ⁿ = 2 at the point (a,b)?

1. n = 1, 2
2. n = 3, 4, -5
3. n = 1, 2, 3
4. n can be any real number

Correct answer: n can be any real number

Solution

Differentiating (x/a)^n+(y/b)^n=2 gives n(x/a)^(n-1)/a + n(y/b)^(n-1) y'/b = 0. At (a,b) this is 1/a + y'/b = 0, so y'=-b/a, and the tangent is x/a+y/b=2 independent of n. Thus n can be any real number.

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