2015 AMC 10A Problem 12

Problem 12 of 25IntermediateAlgebra

Points

(\sqrt{\pi}, a)

and

(\sqrt{\pi}, b)

are distinct points on the graph of

y^2 + x^4 = 2x^2 y + 1.

What is

|a-b|?

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Solution

Substitute

x=\sqrt{\pi}

. Then

x^2=\pi

and

x^4=\pi^2

, so the equation becomes

y^2-2\pi y+\pi^2=1.

Hence

(y-\pi)^2=1

, giving the two possible values

y=\pi+1

and

y=\pi-1

. Their distance is

2

. Thus, C is the correct answer.

Concepts: completing the square · quadratic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.