distance

mms

Jul 2009
67
0
Consider a circle and two points A and B in the exterior of the circle, located in the extension of the diameter of the circle. Determine the path that joins A and B that doesnt touch the interior of the circle and which is the shortest path that doesnt touch the interior of the circle that joins A and B

thanks!
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
Consider a circle and two points A and B in the exterior of the circle, located in the extension of the diameter of the circle. Determine the path that joins A and B that doesnt touch the interior of the circle and which is the shortest path that doesnt touch the interior of the circle that joins A and B

thanks!
A diagram would be useful. But lets suppose A and B are on the same diameter extended but on opposite sides of the circle.

Draw in the tangents from A and B to the circle both on the same side of the diameter. Now the shortest path is along the tangent from A to the point of tangency, then along the circumference of the circle to the point of tangency of the other tangent, then along the tangent to B.

Depending on the exact wording of the question your job is to find the length of this path, or to prove that this is the shortest path.

For the latter you need to consider paths made up of two straight segments from the points to the circle and an arc connecting the points where the lines meet the circle. The fact that the length of an arc of a circle is greater than that of the corresponding chord may be useful.

CB
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Consider a circle and two points A and B in the exterior of the circle, located in the extension of the diameter of the circle. Determine the path that joins A and B that doesnt touch the interior of the circle and which is the shortest path that doesnt touch the interior of the circle that joins A and B

thanks!
The shortest such path is the semi-circle having A and B as endpoints of a diameter.
 

mms

Jul 2009
67
0
For the latter you need to consider paths made up of two straight segments from the points to the circle and an arc connecting the points where the lines meet the circle. The fact that the length of an arc of a circle is greater than that of the corresponding chord may be useful.

CB
i didnt understand this part :( (my english sucks)
also, what if instead of a circle you have an ellipse between the two points? does the same solution still holds?

thanks!
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Actually, I didn't read what you wrote properly. For some reason, I was assuming that A and B are on the circle when you had said clearly that they were exterior to the circle.

Captain Black's response is correct- Draw lines from A and B tangent to the circle. To do that, find the bisector of the line from A to the center of the circle, O, and construct a circle with that point as center and diameter |OA|. That circle will cut the original circle at two points. The line from A to either of those points is gives a tangent to the circle. Do the same with O and B. The straight line from A to the tangent point on the circle, the curve around the circle to the corresponding tangent point for B, then straight to B is the shortest route from A to B that does not go into the interior of the circle.
 

CaptainBlack

MHF Hall of Fame
Nov 2005
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5,271
someplace
i didnt understand this part :( (my english sucks)
also, what if instead of a circle you have an ellipse between the two points? does the same solution still holds?

thanks!
The shortest path will be two line segments from the tangent lines and a part of the boundary of the ellipse.

CB
 
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