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2006 AMC 10A Problems/Problem 11

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Problem

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

\mathrm{(A) \ } \textrm{the\,empty\,set}\qquad \mathrm{(B) \ } \textrm{one\,point}\qquad \mathrm{(C) \ } \textrm{two\,lines} ...

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 \mathrm{(C) \ }.

See also

2006 AMC 10A (ProblemsResources)
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
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