This sounds like something out of the ATPL theory which is based in turn on the pre-war work on gliding theory. You balance the static forces in the descent and derive a formula that says sin gamma = (drag-thrust)/weight. You observe thrust is minimal so say that sin gamma = drag/weight, and then observe that, assuming weight is relatively constant, the smallest value of sin gamma occurs where the drag is least and make the first statement that the most shallow glide angle occurs if you fly at VMD for the weight.
Now hop back to the previous formula that says sin gamma = drag/weight and substitute CD/CL for drag/weight, this leads to the second statement that the glide angle is the lift drag ratio upside down, and this is independent of weight provided you fly at the correct VMD for that weight. Now we have statement #3 which is that, provided you fly at the correct VMD for the weight, the glide angle (flight path angle) will be the same for both a heavy aircraft or a light aircraft and therefore, in answer to the OP's question, the planned track miles will be the same.
Now rate of descent, and for this you need a dynamic diagram with a right angle triangle with descent angle gamma, TAS on the hypotenuse and rate of descent in the vertical. sin gamma = O/H = rate of descent / TAS, therefore rate of descent = Sin gamma x TAS. The heavier aircraft will have the same sin gamma in a VMD glide but IAS will be higher, so TAS will be higher, so rate of descent will be higher. Now we have statement #4 which develops from #3, which is that, provided you fly at the correct VMD for the weight, the glide angle (flight path angle) will be the same for both a heavy aircraft or a light aircraft but the heavier aircraft will have a higher TAS and therefore a higher rate of descent.