Final answer:
The lesion is located in the cortex of the right cerebral hemisphere.
Step-by-step explanation:
The lesion is located in the cortex of the right cerebral hemisphere.
The reason for this is because the patient's eyeballs are turned to the left, which indicates damage to the right cerebral hemisphere's control over eye movement. Additionally, the presence of hemiplegia in the right limbs suggests damage to the right hemisphere, as it controls the left side of the body.