Apache (as proxy) and Node.JS on Ubuntu

My friend asks me to prepare a Node.JS support on his server.

Here is hello.js:

#!/usr/bin/env nodejs
var https = require('https');
var fs = require('fs');

var options = {
  key: fs.readFileSync('/etc/letsencrypt/live/xxxxx/privkey.pem'),
  cert: fs.readFileSync('/etc/letsencrypt/live/xxxxx/cert.pem'),
  ca: fs.readFileSync('/etc/letsencrypt/live/xxxxx/chain.pem')
};

https.createServer(options, function (req, res) {
  res.writeHead(200);
  res.end("Hello world! Node.js is working correctly on HTTPS!!\n");
}).listen(8000);
console.log('Server running at https://127.0.0.1:8080/');

Here is what is added in Appache site configuration:

ProxyRequests Off
ProxyPreserveHost On
ProxyVia Full
<Proxy *>
   Require all granted
</Proxy>

<Location /nodejs>
   ProxyPass https://127.0.0.1:8080
   ProxyPassReverse https://127.0.0.1:8080
</Location>

"sudo node show hello.js" shows that everything is ok. I'm almost sure that problem is in Apache part of this problem. But what?

UPDATED: Some new details. Maybe someone will see a mistake. Apache .conf files:

<VirtualHost *:80>
    ServerAdmin [email protected]
    ServerName example.com
    ServerAlias www.example.com
    DocumentRoot /var/www/example.com
    ErrorLog ${APACHE_LOG_DIR}/error.log
    CustomLog ${APACHE_LOG_DIR}/access.log combined

    RewriteEngine on
    RewriteCond %{SERVER_NAME} =example.com [OR]
    RewriteCond %{SERVER_NAME} =www.example.com
    RewriteRule ^ https://%{SERVER_NAME}%{REQUEST_URI} [END,NE,R=permanent]

    ProxyRequests Off
    ProxyPreserveHost On
    ProxyVia Full
    <Proxy *>
       Require all granted
    </Proxy>

    <Location /nodejs>
       ProxyPreserveHost On
       ProxyPass https://www.example.com:8000
       ProxyPassReverse https://www.example.com:8000
    </Location>

    <Directory "/var/www/example.com">
       AuthType Basic
       AuthName "Restricted Content"
       AuthUserFile /etc/apache2/.htpasswd
       Require valid-user
       Options FollowSymLinks
       AllowOverride All
    </Directory>

</VirtualHost>



********

<IfModule mod_ssl.c>
<VirtualHost *:443>
    ServerAdmin [email protected]
    ServerName example.com
    ServerAlias www.example.com
    DocumentRoot /var/www/example.com
    ErrorLog ${APACHE_LOG_DIR}/error.log
    CustomLog ${APACHE_LOG_DIR}/access.log combined


    Include /etc/letsencrypt/options-ssl-apache.conf
    SSLCertificateFile /etc/letsencrypt/live/example.com/fullchain.pem
    SSLCertificateKeyFile /etc/letsencrypt/live/example.com/privkey.pem

   <Directory "/var/www/example.com">
       AuthType Basic
       AuthName "Restricted Content"
       AuthUserFile /etc/apache2/.htpasswd
       Require valid-user
    </Directory>

</VirtualHost>
</IfModule>

I tried on server and here are two results:

sudo wget --user ime --password geslo https://www.example.com:8000

This WORKS (I get index.html with right response from Node.js app).

sudo wget --user ime --password geslo https://www.example.com/nodejs

This DON'T WORK (I get "nodejs" file with a code for folder contents).

Any idea?