Points (√π, a) and (√π, b) are distinct points on the graph of y^2 + x^4 = 2x^2 y + 1. What is |a-b|?

2015 AMC 10A Problem 12 — Solution

Attempt Problem 12 of the 2015 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

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 $|a-b|?$

$1$

$\dfrac{\pi}{2}$

$2$

$\sqrt{1+\pi}$

$1 + \sqrt{\pi}$

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.