Her husband is ill in . . . . . . . . hospital, so she has to stay at . . . . . . . . home to look after . . . . . . . . children instead of going to work.

Answer: Option A

We use the phrase 'in hospital' when the object of the preposition 'in' (person who's in hospital) is a patient. When a person is in the hospital building, we say 'in the hospital'.
Also, we use the phrase 'in hospital' in British English, whereas the Americans say 'in the hospital'. Since we are following British English, the latter phrase is not preferred. Hence no article is required in the first blank.
'Home' when spoken of in general sense, does not use an article. When the home belongs to the person in it, we say 'at home' and not 'at the home'. In the given sentence, 'home' is spoken of in the general sense and it belongs to the person 'she'.
When the noun is spoken of in general sense, we omit the article. But, when the noun is specific, like 'children' here, we use the definite article 'the'. The noun 'children' is specific as it's made known to the reader that it's the person's children. Hence we use article 'the', and option A is correct.