hughshields on "Modify a Template Tag"

I am trying to modify a template tag to redirect the user authentication link to my Activate page. The current custom registration has the link going to my home page. Here is the existing code:

$url = is_multisite() ? get_blogaddress_by_id( (int) $blog_id) : home_url ('', 'http');

I looked up the home_url template tag in the WP Codex and it looks like this:

<?php home_url( $path, $scheme ); ?>

The $path parameters are: (string) (optional) Path relative to the home URL. Default: None

I believe I want to add the relative page link as the $path but I am getting the syntax wrong or something. I tried

home_url ('activate', 'http');

along with a few other combinations but no luck.