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 == | ||
| − | |||
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>? | ||
| − | <math> \mathrm{(A) | + | <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 == | ||
| + | <cmath> | ||
| + | \begin{align*}(x+y)^2&=x^2+y^2\\ | ||
| + | x^2 + 2xy + y^2 &= x^2 + y^2\\ | ||
| + | 2xy &= 0\end{align*} | ||
| + | </cmath> | ||
| + | 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>. | ||
| + | |||
| + | <asy> draw((0,-50)--(0,50));draw((-50,0)--(50,0));</asy> | ||
== See also == | == See also == | ||
| − | + | {{AMC12 box|year=2006|ab=A|num-b=10|num-a=12}} | |
| + | {{AMC10 box|year=2006|ab=A|num-b=10|num-a=12}} | ||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[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 
?
Solution
Either 
or 
. The union of them is 2 lines, and thus the answer is 
.
![[asy] draw((0,-50)--(0,50));draw((-50,0)--(50,0));[/asy]](http://latex.artofproblemsolving.com/7/7/b/77bc9852606b67fe448031002c3e27a4b8a97bd1.png)
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. 
