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1 - This is the hospital where I was born

2 - This is the hospital whose I was born

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vanni1 - This is the hospital where I was born

Correct, if you add a full stop.

vanni2 - This is the hospital whose I was born

Incorrect. Whose is a possessive form: This is the hospital whose name was changed last year.

CB

Comments  
vanni

1 - This is the hospital where I was born (correct) Where refers to hospital

2 - This is the hospital whose I was born (Wrong relative pronoun)

 Cool Breeze's reply was promoted to an answer.