[RedDevil]

HIDs have a continuous discharge through ionized gas, not unlike CFL tubes.
Just like CFL you need a high voltage spark to get the gas inside the chamber ionized. Once that is done the gas conducts electricity well and only 9 Volt is needed to keep it burning.
The ballasts produce the spark to get the HIDs light up and regulate the voltage to whatever the HID needs when it is lit.

Watts are volts times amps. A 55 Watt lamp running at 12 Volt draws about 4.5 Ampere.
A 35 Watt HID draws just under 4 Ampere at 9 Volt.
During startup it will have a very short jolt at several thousand Volt, but at that voltage it will only draw milliAmps. Once the current starts flowing the voltage drops.

If the ballast just roams off the excess voltage (using a 35W HID at 9V) it will draw 4 amps at 12 Volt, wasting 12 Watt to heat, so the net gain of using HIDs would be minimal compared to halogens: 48 Watt instead of 55.
Ballasts that use a DC-DC converter could draw as little as 38 Watt, assuming 92% efficiency.