2013 AMC 10B Problem 11

Attempt Problem 11 of the 2013 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 10B solutions, or check the answer key.

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11.

Real numbers

x

and

y

satisfy the equation

x^2+y^2=10x-6y-34.

What is

x+y?

1

2

3

6

8

Move all terms to one side and complete two squares

A sum of squares equals zero only when both squares are zero

This can be rewritten as

x^2-10x+y^2+6y+34=0.

Completing the square yields

x^2-10x+25+y^2+6y+9

=(x-5)^2+(y+3)^2

=0.

Since both squared terms are nonnegative and their sum is

0,

both must be

0

. Thus

(x-5)^2 = (y+3)^2=0.

Therefore,

x=5,y=-3,

x+y=2.

Thus, the correct answer is B.

Problem 11 in other years