A levels Maths Circles Question

Have you tried drawing a picture, showing how the circle and the tangents are positioned (roughly)? Once you've drawn a picture, it's pretty obvious the center will not be the mid-point of the two given points.
I haven't looked at the marking scheme yet, but I'd suspect it will use the fact that "tangent is perpendicular to the radius at the point of contact" twice.
Alternatively, your mid-point idea isn't completely useless. This will give you the y-coordinate of the center being 0 (why? there is a theorem about the perpendicular bisector of a chord goes through the center). This saves one calculation as above.

That is how I drew it I’m trying to see how it wouldn't be the centre or how connecting the points isn't the diameter. My bad if I’m being slow😭

I agree, once you draw it out using your knowledge of transformations of graphs and the the fact that a tangent is perpendicular to the radius you should get something that looks like this:
I hope that helps!