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Difference between revisions of "2006 AMC 12A Problems/Problem 11"

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{{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #11]] and [[2006 AMC 10A Problems/Problem 11|2008 AMC 10A #11]]}}
 
== Problem ==
 
== Problem ==
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Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>?
 
Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>?
  
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<math> \mathrm{(A) \ } \;\mathrm{the\; empty\; set}\;\qquad \mathrm{(B) \ } \;\mathrm{one\; point}</math><math>\mathrm{(C) \ } \;\mathrm{two\; lines}\;\qquad \mathrm{(D) \ } \;\mathrm{a\; circle}</math><math>\mathrm{(E) \ } \;\mathrm{the\; entire \; plane}\;</math>
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<math>\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\qquad\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}</math>
  
 
== Solution ==
 
== Solution ==
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<math> (x+y)^2 = x^2 + y^2 </math>
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<cmath>
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\begin{align*}(x+y)^2&=x^2+y^2\\
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<math> x^2 + 2xy + y^2 = x^2 + y^2 </math>
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x^2 + 2xy + y^2 &= x^2 + y^2\\
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2xy &= 0\end{align*}
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<math> 2xy = 0 </math>
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</cmath>
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Either <math> x = 0 </math> or <math> y = 0 </math>. The [[union]] of them is 2 lines, and thus the answer is <math>\mathrm{(C)}</math>.
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Either:
 
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<math> x = 0 </math>
 
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or
 
  
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<math> y = 0 </math>
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<asy> draw((0,-50)--(0,50));draw((-50,0)--(50,0));</asy>
  
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This describes 2 lines <math> \Rightarrow C </math>
 
 
== See also ==
 
== See also ==
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* [[2006 AMC 12A Problems]]
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{{AMC12 box|year=2006|ab=A|num-b=10|num-a=12}}
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*[[2006 AMC 12A Problems/Problem 10|Previous Problem]]
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{{AMC10 box|year=2006|ab=A|num-b=10|num-a=12}}
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*[[2006 AMC 12A Problems/Problem 12|Next Problem]]
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{{MAA Notice}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]

Latest revision as of 16:52, 3 July 2013

The following problem is from both the 2006 AMC 12A #11 and 2008 AMC 10A #11, so both problems redirect to this page.

Problem

Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?

$\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\qquad\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}$

Solution

\begin{align*}(x+y)^2&=x^2+y^2\\ x^2 + 2xy + y^2 &= x^2 + y^2\\ 2xy &= 0\end{align*} Either $x = 0$ or $y = 0$. The union of them is 2 lines, and thus the answer is $\mathrm{(C)}$.

[asy] draw((0,-50)--(0,50));draw((-50,0)--(50,0));[/asy]

See also

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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
2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 10 Problems and Solutions

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

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