Can we show these two circles lie entirely outside each other?

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Prove that the points whose coordinates satisfy the equation \[x^2+y^2+2gx+2fy+c=0\] lie on a circle. State the coordinates of the centre of the circle and the length of its radius.

Prove that the circles \[x^2+y^2-20x-16y+128=0\] and \[4x^2+4y^2+16x-24y-29=0\] lie entirely outside each other, and find the length of the shortest distance from a point on one circle to a point on the other.