2012 AMC 12B Problems/Problem 10
Contents
Problem
What is the area of the polygon whose vertices are the points of intersection of the curves 
and 
Solution 1
The first curve is a circle with radius 
centered at the origin, and the second curve is an ellipse with center 
and end points of 
and 
. Finding points of intersection, we get , 
, and 
, forming a triangle with height of 
and base of 
So the area of this triangle is 
Solution 2
Given the equations and 
, we can substitute 
from the first equation and plug it in to the 2nd equation, giving us 
. After rearranging, 
or 
. The solutions are 
and 
. This gives us the points 
,and . The area of the triangle formed by these points is 
~dragnin (minor edit by KevinChen_Yay)
See Also
| 2012 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
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| All AMC 12 Problems and Solutions | |
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