I have a domain name and a live website. I want to create some subdomains so that they appear as separate websites in Search engines.
For Example take Blogspot. x.blogspot.com and y.blogspot.com are separate websites.
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This is normal behavior. I am not sure what your question is exactly. But here goes.
I am going to assume you are using Apache and host your DNS.
Create the sub-domain on your server however you need. For Apache, this would be almost exactly like any other domains website. You can find the configuration files in /etc/apache2/sites-available or /etc/local/apache2/sites-available. If there is a website already on the web server, you will see the configuration file named something like mydomain.com.conf. Use this file as an example and create your sub-domains. Create a file for each sub-domain like sub-domain.domain.com and cut and paste the contents of your original domain configuration into each. You will need to change some directives.
1] Change your ServerName to something like sub-domain.domain.com to reflect your sub-domain.
2] Remove ServerAlias.
3] Create a root directory for your sub-domain change DocumentRoot and .
4] You may need to change your log file locations as well.
You may have a control panel for this.
Create an entry in your DNS server for the sub-domain. Using the same IP address as the parent domain is fine depending upon your set-up. I am sure it will work okay assuming that all of the sites will be hosted on the same system. You will need an A record for each sub-domain. I cannot tell you how to create the record for your system. I am assuming you will have a control panel for this.
That is it.
Search engines treat all sub-domains websites as separate sites and separate from the parent domain.