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.

Do we know a standard form for the equation of a circle?

Can we rewrite the above equation into this form?

To “prove” the points are on a circle, we need to do more than state a standard equation. Can we show that all the points are the same distance from the centre?

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.

Drawing a diagram will probably help.

Can we use the results from the first part of the question?