Step 1: Fever, cold, cough with a membrane over the tonsils points to diphtheria, caused by Corynebacterium diphtheriae. The aim here is the earliest possible laboratory confirmation.
Step 2: Loeffler's serum slope is the enrichment medium of choice for rapid growth. On it the diphtheria bacilli grow very rapidly, and colonies become visible in just 6 to 8 hours, long before most other contaminating bacteria appear.
Step 3: The other options do not give the earliest diagnosis. L. J. (Lowenstein-Jensen) medium is for Mycobacterium tuberculosis and is very slow growing. MacConkey's agar is a selective medium for enteric gram-negative bacilli. Citrate medium is a biochemical test medium, not for primary isolation of diphtheria.
Step 4: Hence, for the earliest diagnosis the swab should be inoculated on Loeffler's serum slope.