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)} | + | <math>\textbf{(A) } \text{two intersecting parabolas} \qquad \textbf{(B) } \text{two nonintersecting parabolas} \qquad \textbf{(C) } \text{two intersecting circles} \qquad</math> |
− | <math>\textbf{(D)} | + | <math>\textbf{(D) } \text{a circle and a hyperbola} \qquad \textbf{(E) } \text{a circle and two parabolas}</math> |
== Solution == | == Solution == |
Revision as of 14:26, 23 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 |
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