Why did Kauravas and Paandavas considered Drona as their Guru?
What made them to consider Drona as a Guru?
Acharya Drona selecting as a preceptor of Kauravas and Pandavas is mentioned in Mahabharata, Adi Parva, SECTION CXXXIII. The incident is as follows:
Once all princes of Hastinapur are playing with ball and eventually the ball felt in a well. They tried to recover it from well but didn't succeed. There came a Brahmana and recovered that ball by using grass as weapon.
"Drona had thus lived privately for some time in the house of Kripa when one day the heroic princes, all in a company, came out of Hastinapura. And coming out of the city, they began to play with a ball and roam about in gladness of heart. And it so happened that the ball with which they had been playing fell into a well. And thereupon the princes strove their best to recover it from the well. But all the efforts the princes made to recover it proved futile. They then began to eye one another bashfully, and not knowing how to recover it, their anxiety became great. Just at this time they beheld a Brahmana near enough unto them, of darkish hue, decrepit and lean, sanctified by the performance of the Agnihotra and who had finished his daily rites of worship. And beholding that illustrious Brahmana, the princes who had despaired of success surrounded him immediately. Drona (for that Brahmana was no other), seeing the princes unsuccessful, and conscious of his own skill, smiled a little, and addressing them said, 'Shame on your Kshatriya might, and shame also on your skill in arms! You have been born in the race of Bharata! How is it that ye cannot recover the ball (from the bottom of this well)? If ye promise me a dinner today, I will, with these blades of grass, bring up not only the ball ye have lost but this ring also that I now throw down!' Thus saying, Drona that oppressor of foes, taking off his ring, threw it down into the dry well. Then Yudhishthira, the son of Kunti, addressing Drona, said, 'O Brahmana (thou askest for a trifle)! Do thou, with Kripa's permission, obtain of us that which would last thee for life!' Thus addressed, Drona with smiles replied unto the Bharata princes, saying, 'This handful of long grass I would invest, by my mantras, with the virtue of weapons. Behold these blades possess virtues that other weapons, have not! I will, with one of these blades, pierce the ball, and then pierce that blade with another, and that another with a third, and thus shall I, by a chain, bring up the ball.'
"Vaisampayana continued, 'Then Drona did exactly what he had said. And the princes were all amazed and their eyes expanded with delight. And regarding what they had witnessed to be very extraordinary, they said, O learned Brahmana, do thou bring up the ring also without loss of time.'
"Then the illustrious Drona, taking a bow with an arrow, pierced the ring with that arrow and brought it up at once. And taking the ring thus brought up from the well still pierced with his arrow, he coolly gave it to the astonished princes.
Thus all the prices were amazed by the power of Drona and they told this event to Bhishma. Bhishma instantly recognized that only Drona can do this. Then, Bhishma came there and asked Drona the cause of his arrival. And then Drona narrated all the story about His and Drupad's incident where later has insulted Drona for his poverty. And thus, Drona wanted to have great pupils so that he can take revenge to Drupada.
Thus insulted by Drupada, O Bhishma, I have been filled with wrath, I have come to the Kurus, desirous of obtaining intelligent and docile pupils. I come to Hastinapura to gratify thy wishes. O, tell me what I am to do.'
"Vaisampayana continued, 'Thus addressed by the son of Bharadwaja, Bhishma said unto him, 'String thy bow, O Brahmana, and make the Kuru princes accomplished in arms. Worshipped by the Kurus, enjoy with a glad heart to thy fill every comfort in their abode. Thou art the absolute lord, O Brahmana, of what ever wealth the Kurus have and of their sovereignty and kingdom! The Kurus are thine (from this day). Think that as already accomplished which may be in thy heart. Thou art, O Brahmana, obtained by us as the fruit of our great good luck. Indeed, the favour thou hast conferred upon me by thy arrival is great.'