How can I paint a white spot in my area rug?

You can use fabric paint on the carpet.

i used a sharpie ...to make it bleed used rubbing alcohol like the tye dye shirts

watch on youtube under sharpie tye dye

I have used tea to make a white spot disappear.

Is your rug small enough to be dyed? If so my solution would be dye that spot using dye and an appropriate size paint brush, being sure to work where any dye that seeps all the way through will not damage anything. You can set dye by soaking the rug in salt water. In the laundry soap aisle of most stores you'll find a product called "Color Grabbers". Throw one or two in with it the first time you wash it after dyeing, they will prevent any of the dye from remaining in other parts of the rug. They can also be put in with brand new clothing to make sure colors don't bleed onto other clothing.