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 (xβ6)2+(yβ8)2=25?
Answer Choices:
A. 103β
B. 105β
C. 103β+35Οβ
D. 4033ββ
E. 10+5Ο
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 RQβ. Since OP=10,PT=5, and β OTP is a right angle, it follows that β OPT=60β and OT=53β. By similar reasoning, β QPR=60β and QR=53β. Because O,P, and Q are collinear (why?), β RPT=60β, so arc TR is of length 5Ο/3. Hence the length of the shortest route is 2(53β)+35Οβ.
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