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Difference between revisions of "2022 AMC 12B Problems/Problem 8"

(Created page with "== Problem == What is the graph of <math>y^4+1=x^4+2y^2</math> in the coordinate plane? <math>\textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad \textbf{(B)}\ \textbf{T...")
 
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What is the graph of <math>y^4+1=x^4+2y^2</math> in the coordinate plane?
 
What is the graph of <math>y^4+1=x^4+2y^2</math> in the coordinate plane?
  
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<math>\textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad \textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad \textbf{(C)}\ \textbf{Two intersecting circles} \qquad</math>
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<math>\textbf{(A)}\ \text{Two intersecting parabolas} \qquad \textbf{(B)}\ \text{Two nonintersecting parabolas} \qquad \textbf{(C)}\ \text{Two intersecting circles} \qquad</math>
  
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<math>\textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad \textbf{(E)}\ \textbf{A circle and two parabolas}</math>
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<math>\textbf{(D)}\ \text{A circle and a hyperbola} \qquad \textbf{(E)}\ \text{A circle and two parabolas}</math>
  
 
== Solution 1 ==
 
== Solution 1 ==

Revision as of 18:34, 17 November 2022

Problem

What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?

$\textbf{(A)}\ \text{Two intersecting parabolas} \qquad \textbf{(B)}\ \text{Two nonintersecting parabolas} \qquad \textbf{(C)}\ \text{Two intersecting circles} \qquad$

$\textbf{(D)}\ \text{A circle and a hyperbola} \qquad \textbf{(E)}\ \text{A circle and two parabolas}$

Solution 1

Since the equation has even powers of $x$ and $y$, let $y'=y^2$ and $x' = x^2$. Then $y'^2 + 1 = x'^2 + 2y'$. Rearranging gives $y'^2 - 2y' + 1 = x'^2$, or $(y'-1)^2=x'^2$. There are 2 cases: $y' \leq 1$ or $y' > 1$.

If $y' \leq 1$, taking the square root of both sides gives $1 - y' = x'$, and rearranging gives $x' + y' = 1$. Substituting back in $x'=x^2$ and $y'=y^2$ gives us $x^2+y^2=1$, the equation for a circle.

Similarly, if $y' > 1$, we take the square root of both sides to get $y' - 1 = x'$, or $y' - x' = 1$, which is equivalent to $y^2 - x^2 = 1$, a hyperbola. Hence, our answer is $\fbox{A circle and a hyperbola (D)}$, and we're done!

~Bxiao31415

See also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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