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In the xy-plane, the length of the short...

In the xy-plane, 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` is
`10sqrt(3)`
`10sqrt(5)`
`10sqrt(3)+(5pi)/(3)`
`10+5pi`

A

`10sqrt(3)`

B

`10sqrt(5)`

C

`10sqrt(3)+(5pi)/(3)`

D

`10+5pi`

Text Solution

Verified by Experts

The correct Answer is:
C
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 = 90^(@)`, it follows that `/_OPT =60^(@)`and `OT = 5 sqrt(3)`.

Similarly, `PQ = 10` and `PR = 5`
`:. /_QPR = 60^(@)` and `QR = 5sqrt(3)`
Also points O,P and Q are collinear as `m_(OP) = m_(PQ)`
Thus, `/_RPT = 60^(@)` and arc TR is of length `(5pi)/(3)`.
Hence, the length of the shortest route is `2(5sqrt(3))+(5pi)/(3)`.
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