Dought & clear, - [Sacrifices], - Is it permissible for two brothers to share one udhiyah (sacrifice) althoughthey live separately?.

Is it permissible for my brother and I to share in one Udhiyah while
both of us have our own homes in different cities? Our mother lives
sometimes with me, and sometimes with my brother and our father has
passed away. If the mother buys an Udhiyah from her own money, is it
adequate for me and my brother?.
Praise be to Allaah.
Firstly:
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 fromal-Sharh al-Mumti'(7/422).
Secondly:
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 inSaheeh al-Tirmidhi.
It says inTuhfat 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 inZaad 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 inNayl 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.
Thirdly:
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 fromal-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 fromFataawa 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 sacrificetwo horned rams that were white speckled with
blackevery 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 fromMajmoo' Fataawa al-Shaykh Ibn Baaz(18/37).
Fourthly:
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.