Cannot locate content types

Is there a PowerShell script which gives me a list of content types with their located sub site URL in a site collection?

I tried the below script but it only shows me the name of the content types not where they located ( host sub-site URL)

$site = Get-SPSite "My Site URL"

$web = $site.RootWeb

ForEach($ctype in $web.ContentTypes){write-host $ctype.Name": "$ctype.url}

Any suggestion please?