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With a scope tube the cathode is usually at a high -ve potential, often more than -1000V. Since the H-K insulation isn't adequate to withstand that sort of voltage it's usual to connect heater to cathode (either direct or via a resistor) and power it from a separate and highly insulated winding on the mains transformer.
In a TV, the cathode is normally much closer to 0V, though it can be up at 100V or more in some designs. I was working on a professional video monitor yesterday where the CRT heater was supplied by a separate winding for just this reason.
If you have a directly heated CRT, ie no separate cathode, just a filament then you have to have a separate supply unless the filament is grounded at one end. I've not seen a CRT like this but perhaps they exist. Such a filament would normally need a DC supply to avoid hum on the picture.
Direct heating was used in the early days of valves (tubes) with battery supply for the filaments. Also for later battery valves with 2V and 1.4V filaments for portable radios. A number of rectifiers had direct heating from AC; the hum didn't matter here. A separate heater winding was essential for these, and was often used with indrectly heated rectifiers to reduce stress on H-K insulation.
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