How can you tell by looking at the equation of the circle that the center of the circle lies on the y-axis?

Answer 1

The equation for a circle with radius r and centre (X, Y) is given by:

(x - X)² + (y - Y)² = r²

This can be expanded to give:

x² - 2Xx + X² + y² - 2Yy + Y² = r²

→ x² - 2Xx + y² - 2Yy + X² + Y² - r² = 0

If the centre lies on the y-axis, then the x-coordinate of the centre is 0, ie the centre is at (0, Y), then -2X = 0 and the x term disappears to make the equation:

x² + y² -2Yy + Y² - r² = 0

So if the centre lies on the y-axis (x = 0), then there is no term involving just x, only a term involving x².

Answer 2

The maximum and minimum points will be on the y-axis. However, I very much doubt if you could tell the difference if the centre was a microscopic distance from the axis.