Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?                                 [Page No. 95]

Not all radii are possible. In every circle through $A$ and $B$, the segment $AB$ is a chord, and a chord can be at most as long as the diameter. So $2 \times \mathrm{radius} \geq AB$, which means the radius is always at least half of $AB$. Radii smaller than $\frac{AB}{2}$ cannot occur, but every value from $\frac{AB}{2}$ upwards does occur.

Smallest circle: the radius is least when AB itself is a diameter.
Its centre is the midpoint of AB and its radius = $\frac{AB}{2}$.
This is the smallest possible circle through A and B.

Largest circle: there is no largest circle. As the centre moves farther along the perpendicular bisector of AB, the radius grows without limit, so the radius can be made as large as we wish. Hence there is no maximum radius.