AMC12 2013 B

AMC12 2013 B · Q6

AMC12 2013 B · Q6. It mainly tests Quadratic equations, Coordinate geometry.

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x -6y -34$. What is $x + y$?

实数 $x$ 和 $y$ 满足方程 $x^2 + y^2 = 10x -6y -34$。$x + y$ 等于多少？

(A) 1 1

(B) 2 2

(D) 6 6

(E) 8 8

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): By completing the square the equation can be rewritten as follows: $x^2 + y^2 = 10x - 6y - 34,$ $x^2 - 10x + 25 + y^2 + 6y + 9 = 0,$ $(x - 5)^2 + (y + 3)^2 = 0.$ Therefore $x = 5$ and $y = -3$, so $x + y = 2$.

答案（B）：通过配方法，该方程可改写如下： $x^2 + y^2 = 10x - 6y - 34,$ $x^2 - 10x + 25 + y^2 + 6y + 9 = 0,$ $(x - 5)^2 + (y + 3)^2 = 0.$ 因此 $x = 5$ 且 $y = -3$，所以 $x + y = 2$。

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