2006 AMC 10A Problems/Problem 11

Problem

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

$\mathrm{(A) \ } \textrm{the\,empty\,set}\qquad \mathrm{(B) \ } \textrm{one\,point}\qquad \mathrm{(C) \ } \textrm{two\,lines} \qquad \mathrm{(D) \ } \textrm{a\,circle} \qquad \mathrm{(E) \ } \textrm{the\,entire\,plane}$

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 $\mathrm{(C) \ }$ .

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2006 AMC 10A Problems/Problem 11

Problem

Solution

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