why is hydrogen peroxide harmful to the living cell? and if its harmful, then why is it synthesized by the perioxiomes?

The hydrogen peroxide is the by product of the metabolism. It is an oxidising agent and readily oxidises the components of the cell, thus causing severe damage to the cells.  Peroxisomes are microbodies. They are bound by a single membrane and contain enzymes that produce hydrogen peroxide as a by-product. The enzymes of peroxisomes (Peroxidase, another peroxisomal enzyme) use this hydrogen peroxide to oxidize other substrates, including phenols, formic acid, formaldehyde, and alcohol that are toxic to our body by means of the peroxidation reaction.