Does an internal medical examination for a woman affect her fast?
An internal medical examination for a woman invalidates the fast because it involves the entry of a foreign object into the body cavity (jauf) while fasting.
In this case, the woman must refrain from eating and drinking for the rest of the day out of respect for Ramadan and make up for the missed fast after Ramadan.
Such an examination should be avoided in Ramadan and other months unless absolutely necessary, as exposing the private parts is only permissible in cases of necessity.
If necessary, a woman should seek a Muslim female doctor first. If one is unavailable, she may see a female doctor from the People of the Book (Jews and Christians). If neither is available, she may consult a trustworthy and competent Muslim male doctor.
Is it permissible to divide one mudd of fidyah between two people?
No, it is not permissible to divide one mudd between two people, because it would not count as a full meal for either of them.
What is the ruling on swallowing saliva while fasting?
It is permissible for a fasting person to swallow their saliva because avoiding it would cause undue hardship and excessive strictness in religion. Islam discourages such excessiveness since Allah the Almighty intends ease for His followers and does not intend to put them in hardship.
Is it permissible to slaughter a single sheep with the combined intention of both the uḍḥiyyah and the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
According to Shaykh al-Islām Imam Ibn Ḥajar al-Haytamī, it is not permissible to combine the intention of the uḍḥiyyah and the 'aqīqah in a single animal, as each of the two has a distinct and separate cause that differs from the other.
However, Shaykh al-Islām Imam al-Ramlī permitted the combining of both intentions in a single animal — and this position offers a degree of latitude and ease. And Allah Almighty knows best.