Headlight wiring got shorted out?

So, I got smoke coming from the hood when I turned the headlights on. The right headlight wouldn't turn on. I took off the front bumper and the right headlight and found that one of the light bulbs (for the low beam, short distance light) was broken (attachment #1). I also found the actual part that smoked (attachments #2, 3, and 4).

How do I know whether that part still works or I need to buy a new one? Is it ok if I just replace the light bulb and try if it works? I'm kinda worried because these are Xenon light bulbs which are a couple times more expensive than LED bulbs and I don't want to risk damaging one right away. Plus, I don't want to completely destroy the part that smoked.

Thanks for your help
Gio

 

Thank you, xliderider, for your quick response.

Yes, I'm afraid that's a Hylux ballast.

The metal part of the ballast seems to be intact. It's just covered with dirt. It's the rubber cuboid that smoked and melted a little bit.

 

Tombukt2. I'm sorry for my late reply. I got a new job and was busy with orientation the whole week.

You're right about the bulb. It must have been cheap. I couldn't find any dial on the plastic housings of the bulb and couldn't move glass and the bar in either direction. It's just, the bulb seem to have been worn out and the glass tube as well as the white bar moved around instead of being fixed in the plastic base. This might have caused the short circuit because the headlight wouldn't turn on when I was moving the switch on the lever. I've attached a few pictures of that bulb.

I bought even cheaper bulbs, a pair for 11 or so dollars. They seem to be working fine at least for the time being. They don't have the dials either but the glass portion and the bar are fixed tightly in the plastic base. I can't move them around and that makes me feel safer ) Anyway, I'm going to replace those bulbs with LEDs as soon as I can afford.

 

You're right, xliderider. That torched part must have been a capacitor. It contained a battery cell which stinked really badly. I removed it from the ballast and the ballast seems to work just fine now. A clerk at a local store told me to do so. I hope the ballast won't blow up until I replace it with a new one.

I also replaced the damaged bulb. That bulb must have caused the short circuit leaving the circuit without load, I guess. Anyway, thanks again for your helpful replies )