Wonderful Circles that leave us confused

Geometry Level 5

The locus of the mid-point of the chord of circle \(x^{2} + y^{2} = r^{2}\) which subtends a \(90^{o}\) angle at \((p,q)\) lying inside the circle is \(a_{1}x^{2} + a_{2}y^{2} + a_{3}p^{2} + a_{4}q^{2} + a_{5}xp + a_{6}yq = a_{7}r^{a_{8}}\), where \(a_1 = 2.\)

Then find \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} + a_{8}\)

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