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Difference between revisions of "2006 AMC 10A Problems/Problem 11"

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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) \ } \textrm{the\,empty\,set}\qquad \mathrm{(B) \ } \textrm{one\,point}\qquad \mathrm{(C) \ } \textrm{two\,lines} \qquad \mathrm{(D) \ } \textrm{a\,circle} \qquad \mathrm{(E) \ } \textrm{the\,entire\,plane} </math>
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<math> \textbf{(A) } \text{the\,empty\,set}\qquad \textbf{(B) } \textrm{one\,point}\qquad \textbf{(C) } \textrm{two\,lines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire\,plane} </math>
  
 
== Solution ==
 
== Solution ==
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<math>x^2+2xy+y^2=x^2+y^2\Longrightarrow 2xy=0\Longrightarrow xy=0\Longrightarrow x = 0 \textrm{ or } y = 0</math>
 
<math>x^2+2xy+y^2=x^2+y^2\Longrightarrow 2xy=0\Longrightarrow xy=0\Longrightarrow x = 0 \textrm{ or } y = 0</math>
  
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Thus there are two [[line]]s described in this graph, the horizontal line <math>y = 0</math> and the vertical line <math>x=0</math>. Thus, our answer is <math>\mathrm{(C) \ }</math>.
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Thus there are two [[line]]s described in this graph, the horizontal line <math>y = 0</math> and the vertical line <math>x=0</math>. Thus, our answer is <math>\boxed{\textbf{(C) }\textrm{two\,lines}}</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 07:05, 17 December 2021

Problem

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

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

Solution

Expanding the left side, we have

$x^2+2xy+y^2=x^2+y^2\Longrightarrow 2xy=0\Longrightarrow xy=0\Longrightarrow x = 0 \textrm{ or } y = 0$

Thus there are two lines described in this graph, the horizontal line $y = 0$ and the vertical line $x=0$. Thus, our answer is $\boxed{\textbf{(C) }\textrm{two\,lines}}$.

See also

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

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