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Connect your external hard disk to the Mac. Then go to System Preferences > Time Machine and toggle the switch from "Off" to "On."
Then click the "Select Disk…" button to select the drive or volume you want to use for Time Machine(you need to select the connected external disk). Time Machine will ask you if you want to use the disk as your backup destination and will give you the option to encrypt the backups with a password.
Let Time Machine do its work.
Is your hard disk corrupted ? Is it detecting on the computer ?
Can I remote onto your Machine and have a look at your problem? It’s fully secure and you can see what I am doing at all times. If you wish please accept my remote support offer and let me know your time zone and the time period you prefer for remote support. Then I can send you the details to connect.