If your Asus mainboard is less than a few years old, it will most likely have UEFI at it's heart. Also if you can use your mouse within your "BIOS", it will indeed be an UEFI. (It still can be an UEFI if you can't use your mouse though!)
There are ways upon ways to determine if your Windows was installed in UEFI or "Legacy" mode. I'll just share this link http://www.eightforums.com/tutorials/29504-bios-mode-see-if-windows-boot-uefi-legacy-mode.html with you, so you should be able to determine it for Windows. If it is not installed in UEFI mode, the following might not be valid for you!
For nice partitioning you may use the scheme presented in http://www.linuxtechi.com/linux-mint-18-installation-guide-with-screenshots/ . Some additional infos:
- You do not necessarily have to define a seperate
/var partition. Just add it's space requirements to your root
/ partition and only create a bigger root. The installer will then create
/var and in the root partition.
- You may do the same with
/home, though it's relatively common to use a seperate partition. Again, it's your choice.
/swap is said to get double the amount of your RAM. Some say that's a bit anachronistic and unnessecary in times of two-digit GB amounts of RAM. If you have more than 8 GB of RAM, you may use 1.5 times the RAM amount for
/swap, even less the more RAM you have. (Some people even say you can forgo using a
/swap nowadays, but that can get a little risky.)
Now to your special case for
You can create a second seperate EFI partition for your Linux, but you don't have to.
If you don't want to, because you want to use the one from Windows, you may use the drop down menu, which can be seen in your screenshot, to choose the drive where the bootloader should be installed. (I hope I'm not remembering the menu incorrectly. Unfortunately I can not read French.) You would choose
When I was installing Linux Mint as a dual boot next to (UEFI) Windows 10 but on a seperate drive, the Mint installer always used the already present ESP/EFI partition that the Windows installer had created. Even a newly created EFI partition on a seperate drive exclusively for Linux was ignored. I still don't know why.
Fortunately the current Linux Mint installer didn't break my Windows boot.
(Also I could manually move the necessary files to the second EFI partition and everything works like charm now.)
What should happen is, that from then on instead of booting the Windows bootloader, a GRUB instance will load, giving you the possibility to boot Linux or Windows just as you wish. The GRUB and Linux loader will be added to the existing EFI partition.
In your screenshot the entry for
/dev/sdb1 is slightly strange, because usually a FAT variant (most common is FAT32) is used for the EFI-partition. So if
/dev/sdb1 indeed is your EFI partition, Linux might not accept it. Instead it might create a seperate EFI partition on the drive you define in the drop down menu at the bottom. I can't tell the outcome as I have not yet encountered an NTFS formatted EFI partition. This might break your Windows boot temporarily, as the newly installed GRUB might then be unable to find your Windows bootloader. It should be possible to repair this afterwards though.
After you have the two partitions settled in, I recommend using rEFInd for booting the OS of your choice. I found it much nicer to handle than GRUB.