2017 AMC 12B Problem 9

Problem 9 of 25EasierGeometry

A circle has center

(-10, -4)

and radius

13.

Another circle has center

(3, 9)

and radius

\sqrt{65}.

The line passing through the two points of intersection of the two circles has equation

x + y = c.

What is

c?

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Solution

The circles are

(x+10)^2 + (y+4)^2 = 169

and

(x-3)^2 + (y-9)^2 = 65.

Expanding and subtracting the second from the first cancels the

x^2

and

y^2

terms and simplifies to

x + y = 3.

Any intersection point satisfies this, so it is the line through both, and

c = 3.

Thus, the correct answer is A.