3 variables, 3 equations?

Calculus Level 5

Let \(C_1\) and \(C_2\) be defined as: \[\large C_1:x^2+y^2+2\color{red}{g}x+2\color{blue}{f}y+\color{green}{c}=0\] \[(g,f,c \text{ are real parameters})\]

\[\large C_2: y^2-4x+8=0\]

If \(C_1\) passes through the point \((6,3)\) and also cuts \(C_2\) orthogonally at the point with ordinate \(-2\) then \(C_1\) can be written in the form mentioned above, then:

\[\Large \color{red}{g}+\color{blue}{f}-\color{green}{c}=\ ?\]


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