2023 AMC 12B Problem 10

Question

In the

xy

-plane, a circle of radius

4

with center on the positive

x

-axis is tangent to the

y

-axis at the origin, and a circle with radius

10

with center on the positive

y

-axis is tangent to the

x

-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

\dfrac{2}{7}

\dfrac{3}{7}

\dfrac{2}{\sqrt{29}}

\dfrac{1}{\sqrt{29}}

\dfrac{2}{5}

Write both circle equations, then subtract to get the line through both intersection points

The circles simplify to

x^2+y^2=8x

and

x^2+y^2=20y;

set the right sides equal

The circles are

(x-4)^2+y^2=16

and

x^2+(y-10)^2=100,

i.e.

x^2+y^2=8x

and

x^2+y^2=20y.

Subtracting gives

8x=20y,

so the intersection points lie on

y=\tfrac{2}{5}x,

which has slope

\tfrac{2}{5}.

Thus, the correct answer is E.

Problem 10 in other years