This is the 3113th most frequent Korean word.
가족의
"가족의" means "family's" or "of the family" in English.
In this sentence, "가족의" is used to denote possession, specifically the love that belongs to the family.
가족의 사랑은 가슴을 따뜻하게 해요.
The love of family warms the heart.
Here, "가족의" indicates the source or origin of hope, showing connection to the family.
그는 가족의 희망을 위해 열심히 일했어요.
He worked hard for the hope of his family.
In this usage, "가족의" modifies "의미" to describe 'the meaning pertaining to family.'