I believe the problem isn't the voltage itself, which would be found (measured) between gate and ground. The problem is knowing where, in the circuit, the input power is dissipated. The gate-to-ground connection is actually a kind of feedback loop, much like an input-to-gate connection with a resistor between cathode and ground, not decoupled by a capacitor.
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I assume that part of the input power goes, among other things, to the output circuit. Why would this power be dissipated by heating the gate?
