# 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?

Arun
25750 Points
6 years ago
Equation if line joining the two given points is 3y-4x=0 which is satisfied th centre if the circle hence one of the diameters of the given circle will lie on the line joining the origin and (12,16) . Hence the shortest path's distance will be  Distance between origin and given point - length of the line inside the circle(diameter) + half the circumference of the circle (as this will be the path used instead of the diameter)

## Required distance = 20 - 10 + 5*pi = 25.71 units

Ridam
12 Points
6 years ago
Since it cannot be inside the circle it can be tangent to the circle hence the distance can be calculated by joinning 0,0to centre of circle and (12,6 )to to centre as tangent is perpendicular to centre so we can calculte distance using pythagorus theorum on both the triangle and the answer will be 10√3
Ridam
12 Points
6 years ago
The answer is 10√3+5π/3 Since the two distance can be calculated by pythagorus theorum and third distance bycan be calculated by angle at the centre and radius so the ultimately answer is 10√3+5π/3