Apache ProxyPass only if from a Subdomain

I have two servers, both running Apache.
Server 1 and 2 are on the same Public IP address.
Server 1 has a Network IP of 192.168.1.101
Server 2 has a Network IP of 192.168.1.102

example.com goes to Server 1. That is exactly what I want.
I would like for beta.example.com/* to go to Server 2, via an Apache Reverse Proxy.
However, I would like only beta.example.com/* to go to Server 2.

How would I configure Apache to proxy Server 2 only with the one subdomain?

I am assuming that request to public ip goes to server1. You can try this basic configuration on server1 using NameVirtualHost and mod_proxy module.

NameVirtualHost *:80

<VirtualHost *:80>
ServerName beta.example.com
ProxyPass / http://192.168.1.102/
ProxyPassReverse / http://192.168.1.102/
</VirtualHost>

<VirtualHost *:80>
ServerName example.com
DocumentRoot /var/www/example
</VirtualHost>