Is a carbon chiral it two of its groups are cis - trans isomers? For example:

In this image, the two groups sticking to a carbon are cis-trans isomers. So does that make the carbon chiral, exhibiting optical properties? 

What if the two groups were both having the same configuration(both cis or both trans), would it then be achiral?

Thanks for any help :D
Any time all four groups bonded to carbon are distinct, including any sort of isomeric groups, you have a chiral center.  Comments associated with the related question referenced by @Loong indicate that the cis vs trans scenario is even included in the $R$ vs. $S$ naming convention.  This case is an $S$ configuration as the cis group takes second place and is counterclockwise from the hydroxyl group in the appropriate view.