A lakat az ajtón van.

No. In Hungarian, a/az is the definite article, so it means the.

• a lakat = the padlock
• az ajtón = on the door

If you wanted a/an in English, Hungarian usually uses egy.

A and az are just two forms of the same word: the.

• a is used before a consonant sound
• az is used before a vowel sound

So:

• lakat starts with l, so: a lakat
• ajtón starts with a, so: az ajtón

This is similar in idea to English a/an, but in Hungarian it applies to the definite article, not the indefinite one.

Because Hungarian adds a case ending to show location.

The base word is ajtó = door.
Here it becomes ajtón, which means on the door.

The ending here is the superessive case, which often corresponds to English on. So:

• ajtó = door
• ajtón = on the door

Because Hungarian often uses suffixes where English uses prepositions.

English says:

• on the door

Hungarian builds that idea into the noun:

• az ajtón

So instead of a separate word like on, Hungarian attaches a location ending to ajtó.

This is a very common pattern in Hungarian:

• -ban / -ben = in
• -ra / -re = onto
• -on / -en / -ön / -n = on

That is a very common question. Hungarian often does leave out van in the present tense, third person, but not in every kind of sentence.

For example:

• A lakat nagy. = The padlock is big.
  Here van is omitted.

But with location or existence, van is normally kept:

• A lakat az ajtón van. = The padlock is on the door.

So in this sentence, van is needed because the sentence says where something is.

This is a natural word order if you are talking about the padlock and saying where it is.

A simple way to understand it is:

• A lakat = the thing we are talking about
• az ajtón van = the new information about it

So this sentence works well as an answer to:

• Where is the padlock?

Hungarian word order is flexible, but changes in word order often change emphasis.

Yes, that is also grammatical, but the emphasis changes.

• A lakat az ajtón van.
  This starts with the padlock, so it sounds like you are talking about the padlock and saying where it is.

• Az ajtón van a lakat.
  This starts with on the door, so it can sound more like you are highlighting the location, or answering something like What is on the door?

So both are possible, but they are not used in exactly the same situation.

Because both noun phrases are definite:

• a lakat = the padlock
• az ajtón = on the door

In Hungarian, the article still appears even when the noun has a case ending. The ending on ajtón does not replace the article.

So az ajtón is literally built from:

• az = the
• ajtón = on door

Together: on the door

You would usually say:

Van egy lakat az ajtón.

That is different from A lakat az ajtón van.

• Van egy lakat az ajtón. = There is a padlock on the door.
• A lakat az ajtón van. = The padlock is on the door.

The first introduces something as new information.
The second talks about a specific padlock that is already known in the conversation.

You should learn both the base noun and the common case patterns, but not necessarily as completely separate words.

Here is the useful pattern:

• ajtó = door
• ajtón = on the door
• ajtóra = onto the door
• ajtóban = in the door / inside the door as a grammatical pattern

So ajtón is not random. It is the noun ajtó plus a regular location ending. Over time, these endings become one of the most important parts of learning Hungarian grammar.