**Core Concept**
The optimal investigation for planning radiotherapy in esophageal carcinoma involves imaging techniques that provide detailed information about the tumor's extent, relationship to surrounding structures, and potential sites of metastasis. This requires a modality that offers high spatial resolution, multi-planar imaging capabilities, and the ability to assess tumor response to treatment.

**Why the Correct Answer is Right**
The correct answer is **C. CT Scan with Intravenous Contrast**. This modality provides high-resolution images of the esophagus and surrounding tissues, allowing for accurate delineation of the tumor and its relationship to critical structures such as the trachea, bronchi, and major blood vessels. The use of intravenous contrast enhances the visibility of the tumor and surrounding tissues, facilitating precise treatment planning. Furthermore, CT scans can also detect metastatic disease in regional lymph nodes and distant sites, guiding the extent of radiotherapy.

**Why Each Wrong Option is Incorrect**
**Option A:** MRI is not the best choice for planning radiotherapy in esophageal carcinoma due to its limited availability, higher cost, and lower spatial resolution compared to CT scans.

**Option B:** Chest X-ray is not sufficient for planning radiotherapy as it provides limited information about the tumor's extent and relationship to surrounding structures.

**Option D:** PET-CT is useful for assessing tumor metabolism and detecting metastatic disease, but it is not the primary investigation for planning radiotherapy.

**Clinical Pearl / High-Yield Fact**
When planning radiotherapy for esophageal carcinoma, it is essential to use a modality that provides high-resolution images of the tumor and surrounding tissues to ensure accurate treatment delivery and minimize complications.

**Correct Answer: C. CT Scan with Intravenous Contrast**