2006 AMC 10A Problems/Problem 11

Problem

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

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

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 $\boxed{\textbf{(C) }\textrm{two\,lines}}$ .

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

Page

Toolbox

Search

2006 AMC 10A Problems/Problem 11

Problem

Solution

See also