What's Wrong with my iMac?

It's acting very weird -- hoping someone can give me a hand.

When I boot up the machine, the Apple logo comes up -- then turns into a circle with a line through it. Meanwhile, what sounds like a fan in the machine goes into high gear and doesn't stop.

Some background: The machine was getting progressively slower and slower. I did some research and it was suggested that turning off FileVault might help -- so I did that last night.

But when I tried to use the machine today, it seemed even slower than ever -- and was hanging repeatedly. So, I held the on/off switch and it powered down. Rebooted and up came the Apple logo that morphed into the circle with a line through it.

I have a sinking feeling that this is not a good thing ............

Also, back to the slowing down problem -- I'm wondering if one of my kids might have downloaded some sort of huge program from online (Photoshop?) that's been chewing up all the memory. I was thinking about simply deleting each of the three kids accounts (they each have a separate account) -- but will that delete the potential program or other such application that might be the problem?

Thanks for help on this.

Have a look at this:-

Mac OS X: "Broken folder" icon, prohibitory sign, or kernel panic when computer starts

Now try resetting the PRAM and NVRAM before panicking. Keep us posted?