2017 AMC 12B Problem 9

Question

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?

3

3\sqrt{3}

4\sqrt{2}

6

\dfrac{13}{2}

Write both circles as

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

and

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

Subtract one expanded equation from the other; the quadratic terms cancel

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.

Problem 9 in other years