Does Distant Spell work when casting through Familiars?

Question

I have a somewhat peculiar question inasmuch as a sorcerer in a game I am DMing took the 'Magic Initiate' feat because he desperately wanted to be able to cast Find Familiar (a wizard only spell).

A special ability of the familiar is that it can deliver a touch ranged spell cast by the wizard using its reaction, so long as it is within 100 feet of the wizard (effectively extending the wizard's spell range). My mix 'n' match sorcerer, however, has the distant spell metamagic, which allows him to cast touch range spells at a range of 30 feet - he has subsequently asked me whether this would allow him to use his familiar to cast a touch ranged spell at the increased 30 foot range if he were to use his distant spell ability.

The rules are a little unclear as to whether using distant spell would shift a touch spell into a category of 'ranged spell', or if it would remain a touch spell that you simply happen to be casting at range due to the metamagic's effect. Thoughts?

Answer 1

The Metamagic Option, Distant Spell, if used, changes the range of the spell from touch to 30 ft (PHB 102).

Conversely, Find Familiar (PHB 240) gives you the option when you cast a spell with the range of touch to cast it through the familiar, using its reaction to cast the spell.

The two effects are mutually exclusive. If you spend the sorcery point, it is no longer a spell with the range of touch, and can't be pushed through the familiar.