Udhiyah is a confirmed Sunnah (Sunnah mu’akkadah) and is not obligatory, according to the majority of fuqaha’. Some scholars are of the view that it is obligatory for the one who is able to offer it. This is the view of Abu Haneefah and of Ahmad according to one report, and it is the view favoured by Shaykh al-Islam Ibn Taymiyah.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: The view that it is obligatory is stronger that the view that it is not obligatory, but that is subject to the condition that one be able to do it. End quote from al-Sharh al-Mumti’ (7/422).
A sacrifice is valid on behalf of a man and the members of his household, because of the report narrated by al-Tirmidhi (1505) and Ibn Majaah (3147) from ‘Ata’ ibn Yasaar who said: I asked Abu Ayyoob al-Ansaari: How were sacrifices offered among you at the time of the Messenger of Allaah SAWS (peace and blessings of Allaah be upon him)? He said: A man would sacrifice a sheep on behalf of himself and the members of his household, and they would eat some of it and give some to others. Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi.
It says in Tuhfat al-Ahwadhi: This hadeeth clearly states that one sheep is sufficient on behalf of a man and the members of his household, even if they are many, and that is the correct view.
Al-Haafiz Ibn al-Qayyim said in Zaad al-Ma’aad: It was the teaching of the Prophet SAWS (peace and blessings of Allaah be upon him) that one sheep would be sufficient on behalf of a man and the members of his household, even if they were many in number.
Al-Shawkaani said in Nayl al-Awtaar: The correct view is that one sheep is sufficient for the members of a household, even if they are one hundred or more, as is indicated by the Sunnah. End quote.
Members of the household include one's wife and children, and other relatives if they live in the same house and the head of the household spends on them or they share household expenses and food and drink.
But the one who lives in a separate house or has his own income cannot be included in the udhiyah and it is prescribed for him to offer his own sacrifice separately.
Maalik (may Allaah have mercy on him) said, concerning the members of a household who share in the sacrifice: They are the people on whom he spends, whether they are few or many. Muhammad ibn Maalik added: And his child and parents, if they are poor. Ibn Habeeb said: He may include in his sacrifice adult children, even if they are independent of means, and his brother, brother’s son and other relatives if he spends on them, and the members of his household. That is permitted for three reasons: ties of kinship, shared accommodation and spending on him. Muhammad said: He may include his wife in his udhiyah because the wife has the strongest bond with him.
End quote from al-Taaj wa’l-Ikleel Sharh Mukhtasar Khaleel (4/364).
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is it permissible to offer one udhiyah for two brothers living in one house with their children and sharing their food and drink?
He replied: Yes, that is permissible. It is permissible for the members of one household to offer only one sacrifice, even if they are two families, and they will attain thereby the virtue of sacrifice. End quote from Fataawa Noor ‘ala al-Darb.
Shaykh Ibn Baaz (may Allaah have mercy on him) was asked: I am married, praise be to Allaah, and I have children. I live in a city other than the city in which my family live, but on holidays I go to the city where my family are. On Eid al-Adha my children and I came five days before the Eid but we did not offer a sacrifice despite the fact that I am able to, praise be to Allaah.
Is it permissible for me to offer a sacrifice? Is my father’s sacrifice valid on behalf of myself and my wife and children? What is the ruling on sacrifice for the one who is able? Is it obligatory for the one who is not able? Is it permissible to take a loan in order to offer the sacrifice?
He replied: The sacrifice is Sunnah, not obligatory, and one sheep is sufficient on behalf of a man and the members of his household, because the Prophet SAWS (peace and blessings of Allaah be upon him) used to sacrifice two horned rams that were white speckled with black every year, one on behalf of himself and the members of his household, and the other on behalf of those among his ummah who believed in the Oneness of Allaah (Tawheed).
If you live in a separate house, then it is prescribed for you to offer the sacrifice on behalf of yourself and the members of your household, and the sacrifice offered by your father on behalf of himself and the members of his household is not sufficient for you, because you are not living with them in the same house, rather you live in a separate house. There is nothing wrong with a Muslim taking a loan to offer a sacrifice if he is able to repay it. May Allaah help us all.
End quote from Majmoo’ Fataawa al-Shaykh Ibn Baaz (18/37).
Based on the above, your sacrifice is not sufficient for your brother, even if you get together on the days of Eid, and vice versa.
With regard to your mother, her sacrifice is sufficient on behalf of herself and the members of the household with whom she is staying.
And Allaah knows best.