Śrīmad-Bhāgavatam 10.34.30
Blended reading (Sanskrit + English inline)
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anvadhāvat
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He ran after
Imperfect of anu-√dhāv. Same root as v. 27, now singular — Kṛṣṇa alone gives chase. Balarāma remains. Gauḍīya reading: Bhagavān's protective effort is unstinting and personal; He does not delegate the pursuit, He performs it Himself.
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jihīrṣuḥ
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desiring to take away
Desiderative of √hṛ (to seize, take). The verb of the bhakta-king: Bhagavān desires to take the gem — which He will give to Balarāma — and to remove from the world a demon. Both desires are bhakta-centered; Kṛṣṇa wants nothing for Himself.
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śiraḥ-ratnam
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the jewel upon his head
Śiraḥ (head) + ratna (jewel). The brilliant gem the demon wore upon his crest, source of his name. Gauḍīyas read: Kṛṣṇa relieves the demon of every wealth on the way to relieving him of life — but the wealth goes back to the bhakta-Brother, not into hoarding.
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rakṣan striyaḥ
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protecting the women
Rakṣan (present participle of √rakṣ: protecting) + striyaḥ (women). Balarāma's appointed role here. Gauḍīyas note: even when Kṛṣṇa is occupied elsewhere, Vraja is never unguarded — Balarāma stands. The pair-protection is the structure of bhakta-rakṣaṇa.
Śrīla Viśvanātha Cakravartī explains that the women were fatigued from being driven away, and thus Lord Balarāma protected them and consoled them as they rested. Meanwhile Lord Kṛṣṇa went after the demon.
Like time, the inspirer, and death, the inspired, Balarama, the inspirer, said, "Kill him quickly!" and Krsna, the inspired, said, "Yes I will kill him."
“Why did they not throw the śala trees at him from far away and kill him?” Kṛṣṇa wanted to take the jewel, but if he killed the demon, he could not touch the dead body. He desired to take the jewel while the demon was alive. He wanted to give the jewel to Balarāma. That was his intention. Or he wanted to take the jewel because the demon would stay alive as long as the jewel was with him.
Since Kṛṣṇa was the king of Gokula (govindaḥ) it was proper that he pursue the demon in anger to kill him and free the gopīs. “Why did he not kill the demon from far off by throwing stones?” He wanted to take the jewel from the demon when he was alive, since it would be improper to touch him when dead. Balarāma stayed with the women where the demon had left them. He could not override the desire of his younger brother. He was capable of protecting them from the demon since he was strong (balaḥ). Or he was also capable of killing the demon. The reason Kṛṣṇa brought him was so that he could protect the women.
Kṛṣṇa chases the demon, eager for the crest-jewel; Balarāma stays with the women. The Brothers' division of labor is rasa's choreography — vīra and vatsala — and not one gopī is left unguarded while the other Brother sprints. Bhakti's protection is total because Bhagavān always sends Himself in pairs to His own.
My note
Source: vedabase.io