The longest day and the shortest night at these places occur on 21st June.

Explanation:
The Earth revolves around the Sun in an orbit that is slightly tilted on its axis. This tilt is responsible for the changing seasons and the varying lengths of day and night throughout the year.

- Equinox and Solstice:
There are two solstices and two equinoxes that occur in a year. The solstices mark the longest and shortest days of the year, while the equinoxes mark the days when day and night are approximately equal in length.

- Summer Solstice:
The summer solstice occurs around 21st June in the Northern Hemisphere. On this day, the North Pole is tilted towards the Sun at its maximum angle of 23.5 degrees. This results in the Sun appearing at its highest point in the sky and the longest day of the year. The areas near the North Pole experience 24 hours of continuous daylight, while the rest of the Northern Hemisphere enjoys longer days and shorter nights.

- Winter Solstice:
The winter solstice occurs around 22nd December in the Northern Hemisphere. On this day, the North Pole is tilted away from the Sun at its maximum angle of 23.5 degrees. This results in the Sun appearing at its lowest point in the sky and the shortest day of the year. The areas near the North Pole experience 24 hours of continuous darkness, while the rest of the Northern Hemisphere experiences shorter days and longer nights.

- Places near the Equator:
In places near the Equator, such as the tropical regions, the length of day and night remains fairly constant throughout the year. These regions do not experience significant variations in daylight hours.

Conclusion:
Based on the given options, the longest day and the shortest night occur on 21st June, which corresponds to the summer solstice in the Northern Hemisphere.