Praise be to Allah.
Is Udhiyah obligatory?
Udhiyah is a confirmed Sunnah (Sunnah muakkadah) and is not obligatory , according to the majority of jurists. 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 Hanifah and of Ahmad according to one report, and it is the view favoured by Shaykh al-Islam Ibn Taymiyah.
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said: “The view that it is obligatory is stronger than the view that it is not obligatory, but that is subject to the condition that one be able to do it.” (Al-Sharh al-Mumti’, 7/422)
Is one sheep sufficient for the members of a household?
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 Majah (3147) from ‘Ata ibn Yasar who said: I asked Abu Ayyub al-Ansari: How were sacrifices offered among you at the time of the Messenger of Allah (peace and blessings of Allah 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 sahih by al-Albani in Sahih al-Tirmidhi)
It says in Tuhfat al-Ahwadhi: “This hadith 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-Hafiz Ibn al-Qayyim said in Zad al-Ma'ad: “It was the teaching of the Prophet (peace and blessings of Allah 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-Shawkani said in Nayl al-Awtar: “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.”
Can members of the household share in the udhiyah if they live separately?
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.
Malik (may Allah 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 Malik added: And his child and parents, if they are poor. Ibn Habib 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.” (Al-Taj wa’l-Iklil Sharh Mukhtasar Khalil (4/364)
Shaykh Ibn ‘Uthaymin (may Allah 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?
“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.” (Fatawa Nur ‘ala al-Darb)
Shaykh Ibn Baz (may Allah have mercy on him) was asked: I am married, praise be to Allah, and I have children. I live in a city other than the city in which my family lives, 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 Allah.
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?
“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 (peace and blessings of Allah 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 Allah (Tawhid).
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 Allah help us all.” (Majmu’ Fatawa al-Shaykh Ibn Baz (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 Allah knows best.