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>\displaystyle(x+y)^2=x^2+y^2</math>? | Which of the following describes the graph of the equation <math>\displaystyle(x+y)^2=x^2+y^2</math>? | ||
| − | <math> \mathrm{(A) \ } the empty set\qquad \mathrm{(B) \ } one point\qquad \mathrm{(C) \ } two lines\qquad \mathrm{(D) \ } a circle\qquad \mathrm{(E) \ } the entire plane </math> | + | <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> |
== 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</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>\mathrm{(C) \ }</math>. | |
== See Also == | == See Also == | ||
*[[2006 AMC 10A Problems]] | *[[2006 AMC 10A Problems]] | ||
Revision as of 11:23, 31 July 2006
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 
.
