Question

Let $C_1$ and $C_2$ be circles defined by $(x - 10)^2 + y^2 = 36$ and $(x + 15)^2 + y^2 = 81,$ respectively. What is the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q?$

$15$

$18$

$20$

$21$

$24$

Answer: C

Concepts:tangent line similarity Pythagorean Theorem

Difficulty rating: 1660

Solution:

The centers are $A = (10, 0)$ and $B = (-15, 0),$ with radii $6$ and $9,$ so $AB = 25.$ The shortest tangent is the internal one, meeting $\overline{AB}$ at a point $D$ that splits it in the ratio $6 : 9,$ giving $D = (0, 0).$

The right triangles $APD$ and $BQD$ are similar with ratio $2 : 3.$ Then $PD = \sqrt{10^2 - 6^2} = 8$ and $QD = \sqrt{15^2 - 9^2} = 12,$ so $PQ = 8 + 12 = 20.$

Thus, the correct answer is C.

2002 AMC 12A Problem 18

Answer: C

Difficulty rating: 1660

Solution:

Problem 18 in Other Years