A conducting circular loop of radius a is joined to two long,...

The magnetic field at the centre of the loop due to the straight wires is zero because the centre lies on the extended axis of these wires.

The current $i$ divides equally into the two semicircular parts of the loop since they have equal lengths and hence equal resistance. Thus, the current in each semicircle is $i/2$.

The magnetic field at the centre due to the upper semicircle is:
$B_1 = \dfrac{\mu_0 (i/2)}{4a}$ (directed into the plane)

The magnetic field at the centre due to the lower semicircle is:
$B_2 = \dfrac{\mu_0 (i/2)}{4a}$ (directed out of the plane)

Since $B_1$ and $B_2$ are equal in magnitude and opposite in direction, the net magnetic field at the centre is:
$B_{net} = B_1 - B_2 = 0$