Difference between revisions of "2012 AMC 12B Problems/Problem 10"
(→Solution 2) |
(→Solution 2) |
||
Line 11: | Line 11: | ||
==Solution 2== | ==Solution 2== | ||
Given the equations <math>x^2 + y^2 =25</math> and <math>(x-4)^2 + 9y^2 = 81</math>, we can substitute <math>y^2=25-x^2</math> from the first equation and plug it in to the 2nd equation, giving us <math>(x-4)^2+9(25-x^2)=81</math>. After rearranging, <math>8x^2+8x-160=0</math> or <math>x^2+x-20=0</math>. The solutions are <math>x=-5</math> and <math>x=4</math>. This gives us the points <math>(-5,0),(4,3)</math>,and <math>(4,-3)</math>. The area of the triangle formed by these points is <math>27=\fbox{B}</math> | Given the equations <math>x^2 + y^2 =25</math> and <math>(x-4)^2 + 9y^2 = 81</math>, we can substitute <math>y^2=25-x^2</math> from the first equation and plug it in to the 2nd equation, giving us <math>(x-4)^2+9(25-x^2)=81</math>. After rearranging, <math>8x^2+8x-160=0</math> or <math>x^2+x-20=0</math>. The solutions are <math>x=-5</math> and <math>x=4</math>. This gives us the points <math>(-5,0),(4,3)</math>,and <math>(4,-3)</math>. The area of the triangle formed by these points is <math>27=\fbox{B}</math> | ||
− | ~dragnin | + | ~dragnin ~minor edits by KevinChen_Yay |
== See Also == | == See Also == |
Revision as of 22:58, 12 January 2024
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 edits by KevinChen_Yay
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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.