Right Angle's Vertex Is On The Line!

Algebra Level 5

Suppose we have two points \(A(5, 5)\) and \(B(3, 2)\). Point C is placed on the line \(y = 3x+4\) such that \(\triangle ABC\) is a right triangle.

If \(C\) has coordinates \((x, y)\) and the sum of all possible values of \((x, y)\) is \(P\), then find \(11P\).

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