- #1
JDStupi
- 117
- 2
- Homework Statement
- I am trying to work through a problem solving book, and one of the problems is to try to find the shortest path leading from (0,0) to (12,16) that doesn't pass through the circle (x-6)^2+(y-8)^2=25
- Relevant Equations
- (x-6)^2+(y-8)^2=25
and the various line formulas.
This is my attempt at a solution. Point A is the center of the circle (6,8) and Point B is the given point (12,16). I believe that the shortest path would be the one that is equal to the sum of CE and EB or its symmetrical complement. (I forgot to put a point where the top line intersects the y-axis). My problem is 1) To show that this is the case and 2) To find an analytical expression for this.
I must find the slope of a line that makes it from the origin to the point E = (12,y) and lies tangent to the circle at point D. From visual inspection, this point lies to the right of the minimum height of the circle, but how do I determine this optimal point, when I do not know the optimal y for point E?
From there, the solution is a straightforward sum.
I'm not looking for the exact answer, I want coaching on the thinking involved and the process. Thank you all for your help.