Urgent help coordinate geometry please ,!!

Please please can someone help me with the following question. I'm really confused and would really appreciate if someone could provide me with very simple and straighforward steps on how to compete it. I've seen an exam question like this before but I didn't manage to do it and used x axis instead of y. My tuition tutor gave this to my group and I would really love to learn how to complete it.

The circle passes through the points (5,2) and the straight lines y=12 and y=-4 are tangents to the circle . Find the radius and the centre of each of the circles that satisfy these conditions.

Apparently we are supposed to have 2 circle I'm so confused.

All I've got is that 1 circle has a d=16 so a r= 8 ??

So midpoint = 4

And centre = (4,b) ?!

Sketch an x-y plane and sketch the circle in the first and fourth quadrant (that is part of the circle lies in the first quadrant and the other half lies in the fourth quadrant)

Then: Draw the horizontal lines y = 12 and y=-4. You can immediately see that if they are to be tangents, they need to touch the circle at the top and bottom of the circle. So the distance between these two horizontal lines is going to be the diameter: hence diameter = $12 - -4 = 16$ so radius is half of that.

Now the y-coordinate of the circle will be the midpoint of these two lines, hence y=4.

So your circle has equation $(x-a)^2 + (y-4)^2 = 8$

You know that the circle passes through (5,2) - sub those into that equation and find a.

P.S: centre (b,4) not the way you've done it.

Thanks so much

I got a = 3 and a= 7 ?

How what should I do?

Correct. So your final answer is:

Circle 1 has radius 8 and centre (3, 4)
Circle 2 has radius 8 and centre (7, 4)

Ohh thanks so much for your help! I appreciate it so much