2006 AMC 10A Problems/Problem 11

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Problem

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

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

Solution

Expanding the left side, we have

$x^2+2xy+y^2=x^2+y^2\Longrightarrow 2xy=0\Longrightarrow xy=0$

There are two lines described in this graph, horizontal and vertical where x=0 and y=0. Thus, our answer is (C)

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