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2017 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2017 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12B solutions, or check the answer key.

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Concepts:radical axiscirclecoordinate geometry

Difficulty rating: 1410

9.

A circle has center (−10,−4)(-10, -4) and radius 13.13. Another circle has center (3,9)(3, 9) and radius 65.\sqrt{65}. The line passing through the two points of intersection of the two circles has equation x+y=c.x + y = c. What is c?c?

33

333\sqrt{3}

424\sqrt{2}

66

132\dfrac{13}{2}

Solution:

The circles are (x+10)2+(y+4)2=169(x+10)^2 + (y+4)^2 = 169 and (x−3)2+(y−9)2=65.(x-3)^2 + (y-9)^2 = 65. Expanding and subtracting the second from the first cancels the x2x^2 and y2y^2 terms and simplifies to x+y=3.x + y = 3. Any intersection point satisfies this, so it is the line through both, and c=3.c = 3.

Thus, the correct answer is A.

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