# 2015 AMC 10A Problems/Problem 12

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## Problem

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 $|a-b|$? $\textbf{(A)}\ 1 \qquad\textbf{(B)} \ \frac{\pi}{2} \qquad\textbf{(C)} \ 2 \qquad\textbf{(D)} \ \sqrt{1+\pi} \qquad\textbf{(E)} \ 1 + \sqrt{\pi}$

## Solution 1

Since points on the graph make the equation true, substitute $\sqrt{\pi}$ in to the equation and then solve to find $a$ and $b$. $y^2 + \sqrt{\pi}^4 = 2\sqrt{\pi}^2 y + 1$ $y^2 + \pi^2 = 2\pi y + 1$ $y^2 - 2\pi y + \pi^2 = 1$ $(y-\pi)^2 = 1$ $y-\pi = \pm 1$ $y = \pi + 1$ $y = \pi - 1$

There are only two solutions to the equation $(y-\pi)^2 = 1$, so one of them is the value of $a$ and the other is $b$. The order does not matter because of the absolute value sign. $| (\pi + 1) - (\pi - 1) | = 2$

The answer is $\boxed{\textbf{(C) }2}$

## Solution 2

This solution is very closely related to Solution #1 and just simplifies the problem earlier to make it easier. $y^2 + x^4 = 2x^2 y + 1$ can be written as $x^4-2x^2y+y^2=1$. Recognizing that this is a binomial square, simplify this to $(x^2-y)^2=1$. This gives us two equations: $x^2-y=1$ and $x^2-y=-1$.

One of these $y$'s is $a$ and one is $b$. Substituting $\sqrt{\pi}$ for $x$, we get $a=\pi+1$ and $b=\pi-1$.

So, $|a-b|=|(\pi+1)-(\pi-1)|=2$.

The answer is $\boxed{\textbf{(C) }2}$

## Solution 3

This solution is similar to Solution #1 but uses a different way to find $y$ at the end.

Just like Solution #1, we arrive at the conclusion that $y^2 - 2\pi y + \pi^2 = 1$.

Simplifying we get: $y^2 - 2\pi y + \pi^2 -1 = 0$

We now can factor this quadratic. We must find two terms that multiply to $\pi^2 -1$ and add to $2\pi$.

These terms are $\pi+1$ and $\pi-1$.

Subtracting one from the other, we get $2$.

Thus, the answer is $\boxed{\textbf{(C) }2}$

-DuckDuckGooseGoose

## Video Solution

~savannahsolver

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