A circle having center (0, k), with k > 6, is tangent to the lines y = x, y = -x and y = 6. What is the radius of this circle?

2005 AMC 12B Problem 14

Attempt Problem 14 of the 2005 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

A circle having center

(0, k),

with

k \gt 6,

is tangent to the lines

y = x,

y = -x

and

y = 6.

What is the radius of this circle?

6\sqrt{2} - 6

6

6\sqrt{2}

12

6 + 6\sqrt{2}

Tangency to

y = 6

with center

(0,k)

and

k \gt 6

gives

r = k - 6

The distance from

(0,k)

to the line

y = x

\dfrac{k}{\sqrt2},

and it also equals

r

Since the circle is tangent to

y = 6

and its center

(0, k)

is above that line, the radius is

r = k - 6.

The distance from

(0, k)

to the line

x - y = 0

\dfrac{|0 - k|}{\sqrt2} = \dfrac{k}{\sqrt2},

and this must also equal

r.

Setting

\dfrac{k}{\sqrt2} = k - 6

gives

k = \dfrac{6\sqrt2}{\sqrt2 - 1}

= 6\sqrt2\,(\sqrt2+1)

= 12 + 6\sqrt2.

Then

r = k - 6 = 6 + 6\sqrt2.

Thus, the correct answer is E.