2022 AMC 12B Problems/Problem 8
Problem
What is the graph of 
in the coordinate plane?
Solution 1
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 
.
(Solutions 1 and 2 are in essence the same; Solution 1 lets 
for convenience, but the two solutions are otherwise identical.)
Solution 2 (Factoring)
We can subtract 
from both sides and factor the left side to get 
. If we take the square root of both sides, we are left with two equations: 
and 
. In the former case, we get 
, which, is the formula for a hyperbola. This means that a hyperbola will appear in the graph of the equation. In the latter case, we get , which is the equation of a circle which will also appear in the graph. Looking at our options, this is the only valid answer. Hence, the answer is .
~mathman239458
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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