2013 AMC 12B Problem 6

Attempt Problem 6 of the 2013 AMC 12B 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 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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 every term to one side and complete the square in both

x

and

y

A sum of two squares equal to

0

forces each square to be

0

Rearranging gives

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

= 0,

that is

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

Hence

x = 5

and

y = -3,

x + y = 2.

Thus, the correct answer is B.

Problem 6 in other years