The correct option is A he had lost his ball
In reported speech, the word order and pronouns will be different from
the original speech, because the speaker has changed. Accordingly, the
tense will also change, in most cases.
Normally, the tense in the reported speech is one tense back in time from the tense in the direct speech.
The direct speech "I lost my..." is in simple past perfect. Hence while it is being reported, the speech changes to past perfect tense. (one tense back in time)
Option C: 'Had lost' is the past perfect tense structure (had + past participle). The option contains the required form of reported speech. Hence C is correct.
Options A and D: In the direct speech, "I lost my ball..." the subject pronoun 'I' and possessive pronoun 'my' refer to the same person, the speaker: Ajay.
Now the speech is reported by someone else, so the speaker is a different person, and 'I' and 'my' will change its point of view to 'he' and 'his'.
These options contain the pronoun 'my', which do not refer to the original speaker, Ajay. Instead they refer to the new speaker, thereby changing the meaning of the sentence.
Hence A and D are incorrect.
Option B: The tense of the words has not changed. Since the speech has been changed to reported, the tense must be changed to past perfect. Hence B is incorrect.