The charge per unit length of the four quadrant of the ring is $2\mathrm{\lambda },-2\mathrm{\lambda },\mathrm{\lambda }\text{ and }-\mathrm{\lambda }$ respectively. 
The electric field at the center is
 

Formula: $\mathrm{E}=\frac{\mathrm{\lambda }}{2{\mathrm{\pi \epsilon }}_0\mathrm{R}}\sin \frac{\mathrm{\varphi }}{2}$ (along angle bisector)

Intensity of Electric field due to negative charge $=\frac{\mathrm{\lambda }}{4{\mathrm{\pi \epsilon }}_0\mathrm{R}}(-\hat{\mathrm{i}}+\hat{\mathrm{j}})$
Intensity of electric field due to positive charges $=\frac{\mathrm{\lambda }}{4{\mathrm{\pi \epsilon }}_0\mathrm{R}}(-\hat{\mathrm{i}}-\hat{\mathrm{j}})$

Net electric field$=\frac{\mathrm{\lambda }}{2{\mathrm{\pi \epsilon }}_0\mathrm{R}}(-\hat{\mathrm{i}})$