Is it possible to have one ip on main domain and a different ip on a trailing slash domain

Solution 1:

No, because in your example /landing-page is an url-path, which is in this case, simply put, a concept of HTTP protocol.

Your option for a custom DNS is to use a subdomain, e.g.: landing-page.example.com/

You can easily test it in your own machine, simply changing the configuration in hosts file.

Solution 2:

You can implement this with a webserver like nginx that forwards to an upstream webserver any request for /landing-page or any other reverse proxy server, like some types of load balancer that act at the http-level of the network stack.

DNS is mainly used to return just an ip-address, only once that is known and connected with, can be sent an HTTP request like:

GET /landing-page

So no, DNS doesn't help here.