Circle analytic geometry help?

1) Find an eq. for a circle that has its center on the y-axis, that has a radius of 4, and that has only one point on it whose x-coordinate us the same as it's y-coordinate

It would be great if you could help on this one, one of my questions is...since its centered on the y-axis and its radius is 4, as i go up and down the y-axis, wouldn't there be 2 points that have the same value for x and y? (that is including the negative)

What i mean is..for example lets say its centered on (0,3).. wouldnt (-4,3) & (4,3) be on the circle ? maybe im thinking wrong

2) 2 circles are symmetric across the y-axis. The points (7,5) and (3,1) are on endpoints of a diameter of one of the circles. What are the eq.'s of the circles?

Does symmetric mean that they are sticking to each other? I can find the eq. of one of the circles from the endpoints given, but what does them begin symmetric mean, i think a hint here would lead me to the answer

3) A circle with a center at (4,5) is 1st reflected across the y-axis and then across the x-axis. The point (-2,-6) is on this new circle. What are the eq.'s of each of the circles and what is the eq. of the line through the centers of the circles?

I'm lost on this one....