**Core Concept**
The child is presenting with symptoms suggestive of biliary atresia, a condition characterized by the obstruction of bile flow due to the absence or abnormality of the bile ducts. This leads to an accumulation of conjugated (direct) bilirubin, which is a hallmark of this condition.

**Why the Correct Answer is Right**
The most sensitive investigation for diagnosing biliary atresia is the **Hepatobiliary Iminodiacetic Acid (HIDA)** scan, also known as the **Meckel's scan**. This nuclear medicine test evaluates the function and patency of the biliary system. In biliary atresia, the HIDA scan typically shows a failure of the radiotracer to enter the bile ducts, indicating a blockage. This is because the abnormal bile ducts are unable to transport the radiotracer, resulting in a "failed" scan. The HIDA scan is particularly useful in differentiating biliary atresia from other causes of conjugated hyperbilirubinemia.

**Why Each Wrong Option is Incorrect**
**Option A:** Ultrasound is often used to evaluate the liver and bile ducts, but it is not the most sensitive investigation for diagnosing biliary atresia. While ultrasound may show an enlarged liver or bile duct dilation, it cannot reliably distinguish between biliary atresia and other causes of conjugated hyperbilirubinemia.
**Option B:** Liver biopsy may show features suggestive of biliary atresia, such as bile duct proliferation and fibrosis, but it is not the most sensitive investigation for diagnosing this condition.
**Option C:** Endoscopic retrograde cholangiopancreatography (ERCP) is a more invasive procedure that can visualize the bile ducts, but it is not typically used as a first-line investigation for biliary atresia.

**Clinical Pearl / High-Yield Fact**
In biliary atresia, the liver may appear normal on ultrasound, despite significant bile duct obstruction. A high index of suspicion and prompt investigation are essential to diagnose this condition before irreversible liver damage occurs.

**Correct Answer: C. ERCP is not a sensitive test for biliary atresia, which should be diagnosed with a HIDA scan.