What is the shortest distance between two circles, the first having centre $A(5,3)$ and radius 12 , and the other with centre $B(2,-1)$ and radius 6 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
What is the shortest distance between two circles, the first having centre $A(5,3)$ and radius 12 , and the other with centre $B(2,-1)$ and radius 6 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

## Solution

We start by drawing the two circles where the larger circle has centre $A(5,3)$ and the smaller circle has centre $B(2,-1)$. A line is drawn from $A$, through $B$ to meet the circumference of the smaller circle at $C$ and the circumference of the larger circle at $D$. The length $C D$ is the desired length. The length from $A$ to $D$ is given to be 12 and the length from $B$ to $C$ is 6 . We calculate the length of $A B$ to be, $\sqrt{[3-(-1)]^{2}+(5-2)^{2}}=\sqrt{16+9}=5$. To find $C D$, we calculate as follows,

$$
\begin{aligned}
C D & =A D-(A B+B C) \\
& =12-(5+6) \\
& =1
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-248.jpg?height=653&width=613&top_left_y=340&top_left_x=1317)

ANSWER: (A)