Brilli's faithfulness for Billi

Geometry Level 5

Brilli the ant had his ant hill named \(B_1\) which had coordinates \((8,10)\).Brilli had his girlfriend named Billi who had her ant hill named \(B_2\) which had coordinates \((-2,30)\). Once Billi was thirsty and she called Brilli to get her some water from a river named Bribill which had equation \(2x+y-6=0\). Thus to not to displease his girlfriend , Brilli had to take the quickest path which will lead him to Bribill river first and then to Billi's ant hill. Brilli calculated a point \(P\) on the Bribill river such that if he starts from his ant hill , goes to point \(P\) and then goes to Billi's ant hill , he would have taken the shortest possible path requiring less efforts and time.

Find the sum of the coordinates of point \(P\) that Brilli calculated.

Details and Assumptions:

  • Assume the ant-hills as points and the river as a "straight" line.

  • Brilli was always perfect in his calculations.

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