So, Whip is a one-handed martial weapon with reach. Reach is a feature that, among other things, lets you make opportunity attacks with that weapon out to 10 feet instead of just five.

Equipment

Reach (p. 147). This property also
determines your reach for opportunity attacks
with a reach weapon.

— PHB Errata

War Caster lets you cast a spell instead of making an opportunity attack.

Both Hexblade and the Eldritch Invocation Improved Pact Weapon have ways of letting you use a one-handed martial weapon as your spellcasting focus.

So, if you have War Caster, and you have a whip that you can use as a spellcasting focus, would that let you use Eldritch Blast on anyone who provokes an opportunity attack within 10 feet?

As far as I can tell, the real question is whether "spell using the weapon as a spellcasting focus" counts for the "with that weapon" requirement of the Reach property, but I haven't been able to find a clear answer on that.

Yes, but Improved Pact Weapon isn't necessary.

When an enemy moves out of your reach, it provokes an opportunity attack from you, so when an enemy leaves the reach of your whip it provokes an opportunity attack from you (rules on Combat):

You can make an opportunity attack when a hostile creature that you can see moves out of your reach.

If you have the War Caster feat, when an enemy provokes an opportunity attack from you due to movement, you can forego the opportunity attack and instead use your reaction to cast an eligible spell such as eldritch blast (rules on War Caster):

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack.

So, yes, your intended tactic works: the enemy provokes an opportunity attack due to movement, so instead of making an opportunity attack you use your reaction to cast eldritch blast.

However, it's completely irrelevant whether the whip is your spellcasting focus or not. With War Caster, you can provide the somatic components of eldritch blast even if your hands are full of weapons. With Improved Pact Weapon, you could provide the material components of a spell using your pact weapon as your focus, but eldritch blast doesn't have material components, so Improved Pact Weapon isn't necessary to be able to make this tactic work.

All you need is the War Caster feat and a whip.

Yes, all requirements are meant

Let's look at the sequence of action and of the logic here:

1. 
   

   Enemy is at 10' away - They are currently in your active reach if you are wielding a whip because it has the Reach property which extends your reach when using it by 10'.

   
   
   

   
   

   This weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it.

   
   

   
   


2. 
   

   Enemy moves further away - This triggers an Opportunity Attack

   
   
   

   
   

   You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
   The PHB Errata also confirms this:
   This property also determines your reach for opportunity attacks with a reach weapon.

   
   

   
   
   

   At this point, we know that when wielding a whip you have a 10' reach and that OAs are triggered when a creature leaves your reach.

   
   
   

   The condition for reach at 10' has been met and the condition of triggering an OA has been met at 10' when holding a whip.

   
   


3. 
   

   Cast EB - War Caster lets you cast a spell instead of make an opportunity attack.

   
   
   

   
   

   When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

   
   

   
   
   

   This closes the loop on the action. We've covered that the OA is triggered at 10' because of your whip reach, and we've covered that when you get an OA with War Caster you can cast a spell instead. As long as the spellcasting requirements are met (range/components), then you can cast your spell in this case.

   
   

EB's requirements are verbal and somatic. As long as you can speak and have a hand free for your somatic component, then you can cast it.

All conditions have been met (OA at 10' from whip, cast spell instead of attack), so this combo works.

You can cast a spell at a creature only if it leaves your normal reach of 5 feet

Reach The weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it.

You can only provoke an OA with a whip when you are making the OA with the whip.

Using the weapon as a spellcasting focus does not mean you are taking an OA with the whip. The whip is used as part of spellcasting and does not affect the reach of your spell. The reach of your character for OAs made without a whip is simply 5 feet by default.

And since War Caster does not even have you making any sort of OA, simply casting a spell instead of taking an OA, the reach of your whip will always be irrelevant for it, spellcasting focus or no.

Jeremy Crawford agrees in an unofficial tweet:

The War Caster feat relies on your normal reach for the opportunity attack.

This Q&A also comes to the same conclusion in a very related question.