Difference between revisions of "2006 AMC 12A Problems/Problem 11"
(added link to previous and next problem) |
(→See also) |
||
(6 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{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 == | |
− | + | {{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 17:52, 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
.
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.