Let $L\subset L'\in S^3$ be two links such that $L$ has one less number of components than $L'$. Further, $L$ is hyperbolic. Under what conditions is the link $L'$ hyperbolic. To be more specific $L, L'$ are shown in [here](https://i.stack.imgur.com/dbTGW.png).



Hyperbolic links
There is not enough information in your picture to give a definitive answer. If you have more details about the link $L$ then an answer will probably factor through Thurston's characterisation of hyperbolic links. [Purcell's book](http://users.monash.edu/~jpurcell/hypknottheory.html) is an introduction to the subject, and the second half of Theorem 8.17 therein is the precise statement you need.


