Is it permissible to work in a city far from one's sick and elderly mother, while there are minor siblings living with her, and the possibility of studying and working near her?

You are not obligated to live with your mother unless she requests it or is in need of you; in that case, you must stop your studies and stay to care for her. You must show her piety and kindness and strive to please her, as Allah Almighty says: (And your Lord has decreed that you worship none but Him, and to parents, due kindness) [Al-Isra: 23], and His saying: (Give thanks to Me and to your parents; to Me is the final destination) [Luqman: 14]. And it has been narrated in the Hadith: "Stay with her, for Paradise is beneath her feet."