Show that the circle \( x^{2}+y^{2}-(3 p+4) x-(p-2) y+10 p=0 \) passes through the point \( (3,1...

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Show that the circle \( x^{2}+y^{2}-(3 p+4) x-(p-2) y+10 p=0 \) passes through the point \( (3,1) \) for all values of \( p \). If \( p \) varies, find the equation of the locus of the centre of the circle.
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