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Difference between revisions of "2012 AMC 12B Problems/Problem 10"

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~minor edits by [https://artofproblemsolving.com/wiki/index.php/User:Kevinchen_yay KevinChen_Yay]
 
~minor edits by [https://artofproblemsolving.com/wiki/index.php/User:Kevinchen_yay KevinChen_Yay]
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==Solution 3==
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Using our algebra skills we find the points of intersection to be <math>(-5,0),(4,3)</math>,and <math>(4,-3)</math>. Using the shoelace theorem, we can easily find the area of the triangle to be <math>27=\fbox{B}</math>.
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~PeterDoesPhysics
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More information on the shoelace theorem:
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https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem
  
 
== See Also ==
 
== See Also ==

Latest revision as of 08:56, 7 August 2024

Problem

What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 42$

Solution 1

The first curve is a circle with radius $5$ centered at the origin, and the second curve is an ellipse with center $(4,0)$ and end points of $(-5,0)$ and $(13,0)$. Finding points of intersection, we get $(-5,0)$, $(4,3)$, and $(4,-3)$, forming a triangle with height of $9$ and base of $6.$ So the area of this triangle is $9 \cdot 6 \cdot 0.5 =27 \textbf{ (B)}.$

Solution 2

Given the equations $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81$, we can substitute $y^2=25-x^2$ from the first equation and plug it in to the 2nd equation, giving us $(x-4)^2+9(25-x^2)=81$. After rearranging, $8x^2+8x-160=0$ or $x^2+x-20=0$. The solutions are $x=-5$ and $x=4$. This gives us the points $(-5,0),(4,3)$,and $(4,-3)$. The area of the triangle formed by these points is $27=\fbox{B}$

~dragnin

~minor edits by KevinChen_Yay

Solution 3

Using our algebra skills we find the points of intersection to be $(-5,0),(4,3)$,and $(4,-3)$. Using the shoelace theorem, we can easily find the area of the triangle to be $27=\fbox{B}$.

~PeterDoesPhysics

More information on the shoelace theorem: https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem

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

extremely misplaced problem. The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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