Difference between revisions of "2006 AMC 10A Problems/Problem 11"
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== Problem == | == Problem == | ||
| − | Which of the following describes the graph of the equation <math> | + | Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>? |
| − | <math> \ | + | <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 == | ||
Expanding the left side, we have | Expanding the left side, we have | ||
| − | <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> |
| − | 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>\ | + | 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 | + | |
| − | + | == See also == | |
| + | {{AMC10 box|year=2006|ab=A|num-b=10|num-a=12}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 07:05, 17 December 2021
Problem
Which of the following describes the graph of the equation 
?
Solution
Expanding the left side, we have
Thus there are two lines described in this graph, the horizontal line 
and the vertical line 
. Thus, our answer is 
.
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
