IP address of RPi on home Network?

Hi I have added a RPi 4B onto my home network using the eth0 port.
How to determine the IP address is adopts ?
when connected by wlan0 is defaults to a 1982.168.68.xx range, WHY ?
Can I present the range is picks up from the DNS server ?
Home Network is 10.0.0.0/24.
Using arp -a I can see other devices, using nmap still does not show it on 10.0.0.xx network

If you access your modem/router as administrator you will find a heading such as “IP Address Pool”, probably under a menu item like “LAN Settings”. That is where you can assign the range of addresses that devices attached to your LAN can be assigned. Do you have a reason for changing from the default?

At the same place, probably, you will find a panel such as “Client List”. That will give you the names, MAC address and IP address of every attached client. If your device is not listed then that indicates that no connection could be established.

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The pi is headless but using a terminal could login and ran host -I
It is on 192.168.68.104, able to ping from 10.0.0.2, now befuddled by SSH protocols …

You’re going to need to plug in a monitor I think, but as has already been said, the way that most people do it is to log into the Modem/router and find the Wired connected device list and look for it, it will be listed.

However, it all depends on how you have set it up as to what address it has and what range it uses.

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eth0 is connection of the RPi 4B Ethernet port to an Ethernet port on the modem router.
Is this what you eventually want ?
From your post I gather it was not visible to other devices when you did this.

The address is determined by the DHCP settings on the modem/router. Most can be configured to use fixed addresses if you want.

Again, the address is what the DHCP of the modem/router assigns.

The modem/router configuration, looks like it is a Telstra device, most ISPs use 192.168.0.xxx or 192.168.1.xxx for the DHCP range.

DNS server translates URLs to IPs, do you mean DHCP ?
If so the answer is yes, but you need to login to the modem/router to change the settings.

The modem/router handles the connection between the two very different IP addresses.

Your modem/router configuration is what you need to look at and specifically the DHCP area. Unless someone has the same as you it would be hard to provide any further advise. Suggest you contact your ISP and tell them what you want to do, they may or may not help.

Regards
Jim