"This elephant's home is India."
Translation:इस हाथी का घर भारत है।
का/की/के is determined by the gender and number of the following noun. It is का if the noun is masculine and singular, के if masculine and plural and की if it is feminine (singular or plural).
हाथी का घर जंगल में है ('The elephant's home is in the forest') because घर is masculine singular.
हम सब इस हाथी के दोस्त हैं (We are all this elephant's friends) because दोस्त is masculine plural.
उस हाथी की सूंड लंबी है (That elephant's trunk is long) because सूंड is feminine.
There is one more instance when it is के. This is when the noun following it is masculine and singular but is in the oblique case (ie, there is a postposition right after the noun).
Eg: हाथी के घर में घास है (There is grass in the elephant's house) where के is used though घर is masculine and singular because it is in the oblique case due to the postposition में.