To determine the local time at a specific longitude, we need to consider the concept of time zones. The Earth is divided into 24 time zones, each approximately 15 degrees of longitude wide. The standard time of a time zone is typically centered on a specific longitude.

Given that Shillong is located at 92°E longitude, we need to determine the time zone it falls into. The standard meridian for Indian Standard Time (IST) is 82.5°E, which corresponds to the time zone UTC+5:30.

Since Shillong is farther east than the standard meridian for IST, the local time in Shillong will be ahead of IST. The time difference will be determined based on the longitude difference between Shillong and the standard meridian.

The longitude difference between Shillong (92°E) and the standard meridian for IST (82.5°E) is 9.5 degrees. Each degree of longitude corresponds to approximately 4 minutes of time difference. Therefore, the time difference between Shillong and IST is 9.5 degrees * 4 minutes/degree = 38 minutes.

Since the given time is 10.00 a.m. IST, the local time in Shillong would be 10.00 a.m. + 38 minutes = 10.38 a.m.

Therefore, the correct answer is 10.38 a.m.