As I heard, Jatayu was able to fight Ravana and almost won.
How big he was in size and using what power was he able to fight with Ravana? Do any scriptures say about his power?
What are the other activities and role of Jatayu in Ramayana?
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Who was Jatayu?
जटायुरिति मां विद्धि श्येनीपुत्रमरिन्दम।।3.14.32।।
I am born to Aruna and my elder brother is Sampati. O subduer of enemies, know me as Jatayu, son of Syeni.
दशग्रीव स्थितो धर्मे पुराणे सत्यसंश्रयः।
जटायुर्नाम नाम्नाहं गृध्रराजो महाबलः।।3.50.3।।
O tenheaded Ravana, I am Jatayu, king of vultures. I am mighty. An eternal follower, of dharma, I am an adherent to truth.
How big he was in size?
Valmiki Ramayana doesn't mention his size exactly but his body was huge:
अथ पञ्चवटीं गच्छन्नन्तरा रघुनन्दनः।
आससाद महाकायं गृध्रं भीमपराक्रमम्।।3.14.1।।
On the way to Panchavati, Rama, the delight of the Raghu race, met a vulture with a huge body and fearful strength.
ततः पर्वतकूटाभस्तीक्ष्णतुण्डः खगोत्तमः।
वनस्पतिगतश्श्रीमान्व्याजहार शुभां गिरम्।।3.50.2।।
Glorious Jatayu, the best among the birds, looked like the peak of a mountain. His beak was sharp. His words were auspious. He spoke from the top of the tree.
Using what power he was able to fight with Ravana?
विददार नखैरस्य तुण्डं पृष्ठे समर्पयन्।।3.51.35।।
Jatayu who had his claws, wings and beak as the only weapons pierced and pricked Ravana all over with his nails by pressing his beak on his back and pulled out his hair.
तस्य तीक्ष्णनखाभ्यां तु चरणाभ्यां महाबलः। चकार बहुधा गात्रे व्रणान्पतगसत्तमः।।3.51.7।।
The mighty, virtuous vulture with his sharp claws wounded many parts of the body of Ravana.
स तानि शरवर्षाणि पक्षाभ्यां च विधूयच।
चरणाभ्यां महातेजा बभञ्जास्य महद्धनुः।।3.51.14।।
Powerful Jatayu scattered the shower of arrows with his wings and broke down the great bow with his claws.
What other activities of Jatayu are mentioned in Ramayana?
स तौ मधुरया वाचा सौम्यया प्रीणयन्निव।
उवाच वत्स मां विद्धि वयस्यं पितुरात्मनः।।3.14.3।।
With a sweet, pleasing, and gentle voice he said to both of them, O dear know me to be a friend of your father.
जटायुषं तं प्रतिपूज्य राघवो मुदा परिष्वज्य च सन्नतोऽभवत्।
पितुर्हि शुश्राव सखित्वमात्मवान् जटायुषा सङ्कथितं पुनः पुनः।।3.14.34।।
Rama worshipped Jatayu and bending down, embraced him. He heard from him again and again several events relating to Jatayu's friendship with his father.
पुरा वृत्रवधे वृत्ते परस्परजयैषिणौ।
आदित्यमुपयातौ स्वो ज्वलन्तं रश्मिमालिनम्।।4.58.4।।
'In olden days, when Indra killed Vrutra we (Jatayu and I (Jatayu's elder brother, the glorious Sampati)) seeking to challenge him reached the Sun blazing with a garland of beams.
Translation Source: Valmiki-IITK portal