In Flame Emission Spectroscopy, what part of the flame is used to measure the concentration of the analyte?
I have read the article but it was never mentioned what part of the flame does the transducer or sensor was placed on.


It was never mentioned because no sensor or transducer is actually placed on or in the flame. What spectroscopist do is that they focus the light (coming from the flame) into a device called the monochromator. The job of the monochromator is to separate the wavelengths of the light from the flame. The desired wavelength is then focussed on the detector. Thus the detector is really far from the actual flame.

Now the key question is which part of the flame is useful. There is no single correct answer. The criterion is we scan the height of the flame and check which part of the flame is showing the maximum intensity of the desired signal. This is done by changing the height of the burner. Most spectrometers do it automatically but old lovely instruments had knobs for manual adjustment.

Normally for most elements one would like to see the hottest part of the flame, where the concentration of the atomic emission is the highest.