First off, the pagefile is not virtual memory. Virtual memory uses the hard drive (most commonly) as additional RAM. So if you had 1GB of RAM and 1GB of virtual memory, you would have 2GB of memory. A pagefile is where the operating system moves the least used pages of memory (hence, pagefile) out to the hard disk free up more RAM for the applications that you are actually using. A pagefile does not give you more RAM, it just used the RAM you have more efficiently.
Now to answer your question of need to put the pagefile on your system partion: No.
... And yes.
If your system is configured for crash dumps, you'll need to have pagefile on your system partition, or Windows won't be able to write out the process memory in the event of a crash. I dont know if there it is a fixed amount, or a factor of your physical memory, but from what I have seen, it is very small. If you try this, Windows will tell you what the minimum should be. That is the only need for putting your pagefile on your system partition - and it isnt mandatory. There are people who run Windows without a pagefile, however many people (including Microsoft) think that it is a bad idea.
You could have a small pagefile on your C: partition and a larger one on your D:, as you can spread your pagefile across multiple partitions. However, that doesnt answer your question.
Should you put the pagefile on the system partion? Most probably.
Modern hard drives, with zone recording, the outermost track is the fastest part of the drive. And writing begins at the outermost track. Unless you did something odd with creating your partitions, the C: is your system partition and it is the first partition and outermost track.
Why does this matter? RAM is one of the fastest components of a PC and hard drives are one of the slowest. Memory that is being paged is being read from and written to the disk. You want that to be as fast as possible. Everyone has seen a computer grind to a halt and the you look at the hard drive light, seeing it constantly on. That is due to the computer not having enough memory and the system is paging non-stop. Your might have enough RAM and your hard drive might be fast enough that this issue might not be noticeable... But I have seen it happen on the beefiest machines.
So there are a handful of choices to consider, but in the end, you really can do whatever you want.
However, since you asked, I say put it on your C:.