**Core Concept**
The question tests the understanding of the neuroanatomical localization of a stroke lesion based on the clinical presentation of a patient with right upper motor neuron paralysis of the facial nerve and hemiplegia. The facial nerve is a branch of the cranial nerve VII, which emerges from the lateral surface of the pons in the brainstem.

**Why the Correct Answer is Right**
The patient's symptoms suggest a lesion in the brainstem or the internal capsule. The upper motor neuron paralysis of the facial nerve indicates a lesion in the upper motor neuron pathway of the facial nerve, which arises from the motor cortex and descends through the internal capsule and the pons. The hemiplegia on the right side, with equal paralysis of the arm and leg, suggests a lesion in the corticospinal tract, which also descends through the internal capsule and the brainstem.

**Why Each Wrong Option is Incorrect**
**Option A:** A lesion in the left hemisphere would result in left-sided hemiplegia, not right-sided.
**Option B:** A lesion in the cerebellum would result in ataxia, not upper motor neuron paralysis of the facial nerve.
**Option C:** A lesion in the spinal cord would result in lower motor neuron paralysis, not upper motor neuron paralysis.

**Clinical Pearl / High-Yield Fact**
Remember that the corticospinal tract and the facial nerve are both located in the internal capsule and the brainstem, and lesions in these areas can result in a combination of upper motor neuron paralysis of the facial nerve and hemiplegia.

**Correct Answer:** D. The internal capsule.