Step 1: Two findings must coexist: ketone bodies present, but the urine free of glucose. We screen each option against that pair.
Step 2: Uncontrolled diabetes mellitus certainly makes ketones, however the high blood glucose overflows into urine, so glycosuria is the rule. That breaks the condition and rules out option a.
Step 3: Diabetes insipidus is purely an ADH or renal water-handling problem producing dilute polyuria; it neither drives ketosis nor causes glycosuria. Obesity by itself also fails to generate meaningful ketosis. Both are out.
Step 4: During prolonged fasting, insulin falls and fatty acids flood the liver to be turned into ketone bodies, while gluconeogenesis keeps blood sugar low to normal. Result: ketosis with Rothera positive but Benedict negative urine, exactly ketosis without glycosuria.
\[\boxed{\text{Starvation}}\]