Problem:
In the xy-plane, what is the length of the shortest path from (0,0) to (12,16) that does not go inside the circle (x−6)2+(y−8)2=25?
Answer Choices:
A. 103​
B. 105​
C. 103​+35π​
D. 4033​​
E. 10+5Ï€
Solution:
Let O=(0,0),P=(6,8), and Q=(12,16). As shown in the figure, the shortest route consists of tangent OT, minor arc TR, and tangent RQ​. Since OP=10,PT=5, and ∠OTP is a right angle, it follows that ∠OPT=60∘ and OT=53​. By similar reasoning, ∠QPR=60∘ and QR=53​. Because O,P, and Q are collinear (why?), ∠RPT=60∘, so arc TR is of length 5π/3. Hence the length of the shortest route is 2(53​)+35π​.
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