Difference between revisions of "2022 AMC 12B Problems/Problem 8"
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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? | ||
| − | <math>\textbf{(A) } \text{two intersecting parabolas} \qquad \textbf{(B) } \text{two nonintersecting parabolas} \qquad \textbf{(C) } \text{two intersecting circles} \qquad | + | <math>\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}</math> |
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== Solution == | == Solution == | ||
Revision as of 02:36, 26 November 2023
Problem
What is the graph of 
in the coordinate plane?
Solution
Since the equation has even powers of 
and 
, let 
and 
. Then 
. Rearranging gives 
, or 
. There are two cases: 
or 
.
If , taking the square root of both sides gives 
, and rearranging gives 
. Substituting back in 
and gives us 
, the equation for a circle.
Similarly, if , we take the square root of both sides to get 
, or 
, which is equivalent to 
, a hyperbola.
Hence, our answer is 
.
Video Solution(1-16)
~~Hayabusa1
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
