Q.ys In the xy-plane, the length of the shortest path from (0.0) to (12.16) that does not go inside the circle 6) (y-8) 25 is (D) 10 (B) 10 5 (A) 10 Dps' on Circle

Options

A. `10sqrt(3)`
B. `10sqrt(5)`
C. `10sqrt(3)+(5pi)/(3)`
D. `10+5pi`

Answer

C

Explanation

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)`.