Suggestions for dropping resistor on LM78 regulator
A resistor can be used in two very different ways to reduce heat load on a three-pin regulator chip:



simulate this circuit &ndash; Schematic created using CircuitLab
The second circuit cannot be used in your application because the regulator chip has a maximum input voltage of 35V and can't handle 44V.
The first circuit can be used only if the load is fairly constant. If the load is switched on and off, this circuit can't be used, because when off, very little current flows, and input voltage rises so that the zener diode must sink all the current, and likely overheats. In most cases, the zener is only included to protect the regulator during very short no-current periods...the regulator cannot withstand an input over-voltage higher than 35V, even for an instant.
Rseries reduces heat load on the 7812 regulator by reducing input voltage. 7812 dropout voltage is 2V, so input should be no lower than +14V. In the example circuit shown above, with a constant 0.4A load current flowing, and constant 44V input voltage, Rseries should be no higher than (44-14)/0.4 = 75 ohms. In a practical application, neither 44V input voltage, nor Iload of 0.4A will be exactly constant (44V input will have 50/60Hz ripple for example). To the extent that these two are constant, this circuit is very useful. Where both are somewhat variable, Rload must be made smaller than 75 ohms, and the regulator will run hotter.
Choosing Rseries value requires you to know how much load current flows, how much it can vary. You must design Rseries using maximum load current, so that regulator input remains above 14V. You must also know how much 50/60Hz ripple is on the DC input voltage, so you can know its minimum peak voltage. AC line voltage variation is also involved. You would choose Rseries so that regulator input voltage at no time falls below 14V, and at no time rises above 33V (except perhaps for less than a millisecond).