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For the curve xy = c², suppose the normal drawn at the point corresponding to parameter t1 intersects the curve again at parameter t2. Then which relation is true?

1. t1³ t2 = 1
2. t1³ t2 = −1
3. t1 t2³ = −1
4. t1 t2³ = 1

Correct answer: t1³ t2 = −1

Solution

Parametrize xy=c^2 as (ct, c/t). Slope of curve is -1/t^2, so the normal at t1 has slope t1^2. Setting the normal line equal to the curve again gives t2=-1/t1^3, hence t1^3 t2 = -1.

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