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For the point (a,0), the greatest possible distance from the curve 2x²+y²-2x=0 is

1. √(1-2a+a²)
2. √(1+2a+2a²)
3. √(1+2a-a²)
4. √(1-2a+2a²)

Correct answer: √(1-2a+2a²)

Solution

From 2x^2+y^2-2x=0, y^2=2x-2x^2 with 0<=x<=1. Distance^2 from (a,0): (x-a)^2+y^2 = -x^2+(2-2a)x+a^2, maximized at x=1-a giving (1-a)^2+a^2 = 1-2a+2a^2. So greatest distance = sqrt(1-2a+2a^2).

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