If $I_1=I_2$ then $\overset{-}{B}$ cannot be equal to zero at the origin $(0,0,0)$
If $I_1 > 0$ and $I_2 < 0$ then $\overset{-}{B}$ can be equal to zero at the origin $(0,0,0)$
If $I_1 < 0$ and $I_2 > 0$ then $\overset{-}{B}$ can be equal to zero at the origin $(0,0,0)$
If $I_1=I_2$ ,then the z-component of the magnetic field at the centre of the loop is $\left(-\frac{\mu_0 I}{2R}\right)$

Solution:

A) At origin,

\overset{-}{B}=0

due to two wries if

I_1=I_2

,hence

\overset{-}{B}_\text{net}

at origin is equal to

\overset{-}{B}

due to ring, which is non-zero

B) If

I_1 > 0

and

I_2 < 0,\overset{-}{B}

at origin due to wires will be along

+\overset{-}{k}

direction and

\overset{-}{B}

due to ring is along

-\overset{-}{k}

direction and hence

\overset{-}{B}

can be zero at origin.

C) If

I_1 < 0

and

I_2 > 0

,at origin due to wires along

-\overset{-}{k}

and also along

-\overset{-}{k}

due to ring , hence

\overset{-}{B}

cannot be zero

D) At centre of ring,

\overset{-}{B}

due to wires is along x-axis.

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