To set the stage for the question please read 2 Kings 17 followed by John 4 first, then answer the question with these scriptures fresh in mind.
1In the twelfth year of Ahaz king of Judah began Hoshea the son of Elah to reign in Samaria over Israel nine years. 2And he did that which was evil in the sight of the LORD, but not as the kings of Israel that were before him. 3Against him came up Shalmaneser king of Assyria; and Hoshea became his servant, and gave him presents. 4And the king of Assyria found conspiracy in Hoshea: for he had sent messengers to So king of Egypt, and brought no present to the king of Assyria, as he had done year by year: therefore the king of Assyria shut him up, and bound him in prison.
5Then the king of Assyria came up throughout all the land, and went up to Samaria, and besieged it three years.
6In the ninth year of Hoshea the king of Assyria took Samaria, and carried Israel away into Assyria, and placed them in Halah and in Habor by the river of Gozan, and in the cities of the Medes.
7For so it was, that the children of Israel had sinned against the LORD their God, which had brought them up out of the land of Egypt, from under the hand of Pharaoh king of Egypt, and had feared other gods, 8And walked in the statutes of the heathen, whom the LORD cast out from before the children of Israel, and of the kings of Israel, which they had made. 9And the children of Israel did secretly those things that were not right against the LORD their God, and they built them high places in all their cities, from the tower of the watchmen to the fenced city. 10And they set them up images and groves in every high hill, and under every green tree: 11And there they burnt incense in all the high places, as did the heathen whom the LORD carried away before them; and wrought wicked things to provoke the LORD to anger: 12For they served idols, whereof the LORD had said unto them, Ye shall not do this thing. 13Yet the LORD testified against Israel, and against Judah, by all the prophets, and by all the seers, saying, Turn ye from your evil ways, and keep my commandments and my statutes, according to all the law which I commanded your fathers, and which I sent to you by my servants the prophets. 14Notwithstanding they would not hear, but hardened their necks, like to the neck of their fathers, that did not believe in the LORD their God. 15And they rejected his statutes, and his covenant that he made with their fathers, and his testimonies which he testified against them; and they followed vanity, and became vain, and went after the heathen that were round about them, concerning whom the LORD had charged them, that they should not do like them. 16And they left all the commandments of the LORD their God, and made them molten images, even two calves, and made a grove, and worshipped all the host of heaven, and served Baal. 17And they caused their sons and their daughters to pass through the fire, and used divination and enchantments, and sold themselves to do evil in the sight of the LORD, to provoke him to anger. 18Therefore the LORD was very angry with Israel, and removed them out of his sight: there was none left but the tribe of Judah only.
19Also Judah kept not the commandments of the LORD their God, but walked in the statutes of Israel which they made. 20And the LORD rejected all the seed of Israel, and afflicted them, and delivered them into the hand of spoilers, until he had cast them out of his sight.
21For he rent Israel from the house of David; and they made Jeroboam the son of Nebat king: and Jeroboam drave Israel from following the LORD, and made them sin a great sin. 22For the children of Israel walked in all the sins of Jeroboam which he did; they departed not from them; 23Until the LORD removed Israel out of his sight, as he had said by all his servants the prophets. So was Israel carried away out of their own land to Assyria unto this day.
24And the king of Assyria brought men from Babylon, and from Cuthah, and from Ava, and from Hamath, and from Sepharvaim, and placed them in the cities of Samaria instead of the children of Israel: and they possessed Samaria, and dwelt in the cities thereof. 25And so it was at the beginning of their dwelling there, that they feared not the LORD: therefore the LORD sent lions among them, which slew some of them. 26Wherefore they spake to the king of Assyria, saying, The nations which thou hast removed, and placed in the cities of Samaria, know not the manner of the God of the land: therefore he hath sent lions among them, and, behold, they slay them, because they know not the manner of the God of the land.
27Then the king of Assyria commanded, saying, Carry thither one of the priests whom ye brought from thence; and let them go and dwell there, and let him teach them the manner of the God of the land. 28Then one of the priests whom they had carried away from Samaria came and dwelt in Bethel, and taught them how they should fear the LORD.
29Howbeit every nation made gods of their own, and put them in the houses of the high places which the Samaritans had made, every nation in their cities wherein they dwelt. 30And the men of Babylon made Succothbenoth, and the men of Cuth made Nergal, and the men of Hamath made Ashima, 31And the Avites made Nibhaz and Tartak, and the Sepharvites burnt their children in fire to Adrammelech and Anammelech, the gods of Sepharvaim. 32So they feared the LORD, and made unto themselves of the lowest of them priests of the high places, which sacrificed for them in the houses of the high places. 33They feared the LORD, and served their own gods, after the manner of the nations whom they carried away from thence.
34Unto this day they do after the former manners: they fear not the LORD, neither do they after their statutes, or after their ordinances, or after the law and commandment which the LORD commanded the children of Jacob, whom he named Israel; 35With whom the LORD had made a covenant, and charged them, saying, Ye shall not fear other gods, nor bow yourselves to them, nor serve them, nor sacrifice to them: 36But the LORD, who brought you up out of the land of Egypt with great power and a stretched out arm, him shall ye fear, and him shall ye worship, and to him shall ye do sacrifice. 37And the statutes, and the ordinances, and the law, and the commandment, which he wrote for you, ye shall observe to do for evermore; and ye shall not fear other gods. 38And the covenant that I have made with you ye shall not forget; neither shall ye fear other gods. 39But the LORD your God ye shall fear; and he shall deliver you out of the hand of all your enemies. 40Howbeit they did not hearken, but they did after their former manner. 41So these nations feared the LORD, and served their graven images, both their children, and their children's children: as did their fathers, so do they unto this day.
1When therefore the Lord knew how the Pharisees had heard that Jesus made and baptized more disciples than John, 2(Though Jesus himself baptized not, but his disciples,) 3He left Judaea, and departed again into Galilee. 4And he must needs go through Samaria. 5Then cometh he to a city of Samaria, which is called Sychar, near to the parcel of ground that Jacob gave to his son Joseph. 6Now Jacob's well was there. Jesus therefore, being wearied with his journey, sat thus on the well: and it was about the sixth hour.
7There cometh a woman of Samaria to draw water: Jesus saith unto her, Give me to drink. 8(For his disciples were gone away unto the city to buy meat.) 9Then saith the woman of Samaria unto him, How is it that thou, being a Jew, askest drink of me, which am a woman of Samaria? for the Jews have no dealings with the Samaritans. 10Jesus answered and said unto her, If thou knewest the gift of God, and who it is that saith to thee, Give me to drink; thou wouldest have asked of him, and he would have given thee living water. 11The woman saith unto him, Sir, thou hast nothing to draw with, and the well is deep: from whence then hast thou that living water? 12Art thou greater than our father Jacob, which gave us the well, and drank thereof himself, and his children, and his cattle? 13Jesus answered and said unto her, Whosoever drinketh of this water shall thirst again: 14But whosoever drinketh of the water that I shall give him shall never thirst; but the water that I shall give him shall be in him a well of water springing up into everlasting life.
15The woman saith unto him, Sir, give me this water, that I thirst not, neither come hither to draw. 16Jesus saith unto her, Go, call thy husband, and come hither. 17The woman answered and said, I have no husband. Jesus said unto her, Thou hast well said, I have no husband: 18For thou hast had five husbands; and he whom thou now hast is not thy husband: in that saidst thou truly. 19The woman saith unto him, Sir, I perceive that thou art a prophet. 20Our fathers worshipped in this mountain; and ye say, that in Jerusalem is the place where men ought to worship. 21Jesus saith unto her, Woman, believe me, the hour cometh, when ye shall neither in this mountain, nor yet at Jerusalem, worship the Father. 22Ye worship ye know not what: we know what we worship: for salvation is of the Jews. 23But the hour cometh, and now is, when the true worshippers shall worship the Father in spirit and in truth: for the Father seeketh such to worship him. 24God is a Spirit: and they that worship him must worship him in spirit and in truth. 25The woman saith unto him, I know that Messias cometh, which is called Christ: when he is come, he will tell us all things. 26Jesus saith unto her, I that speak unto thee am he.
27And upon this came his disciples, and marvelled that he talked with the woman: yet no man said, What seekest thou? or, Why talkest thou with her? 28The woman then left her waterpot, and went her way into the city, and saith to the men, 29Come, see a man, which told me all things that ever I did: is not this the Christ? 30Then they went out of the city, and came unto him.
31In the mean while his disciples prayed him, saying, Master, eat. 32But he said unto them, I have meat to eat that ye know not of. 33Therefore said the disciples one to another, Hath any man brought him ought to eat? 34Jesus saith unto them, My meat is to do the will of him that sent me, and to finish his work. 35Say not ye, There are yet four months, and then cometh harvest? behold, I say unto you, Lift up your eyes, and look on the fields; for they are white already to harvest. 36And he that reapeth receiveth wages, and gathereth fruit unto life eternal: that both he that soweth and he that reapeth may rejoice together. 37And herein is that saying true, One soweth, and another reapeth. 38I sent you to reap that whereon ye bestowed no labour: other men laboured, and ye are entered into their labours.
39And many of the Samaritans of that city believed on him for the saying of the woman, which testified, He told me all that ever I did. 40So when the Samaritans were come unto him, they besought him that he would tarry with them: and he abode there two days. 41And many more believed because of his own word; 42And said unto the woman, Now we believe, not because of thy saying: for we have heard him ourselves, and know that this is indeed the Christ, the Saviour of the world.
43Now after two days he departed thence, and went into Galilee. 44For Jesus himself testified, that a prophet hath no honour in his own country. 45Then when he was come into Galilee, the Galilaeans received him, having seen all the things that he did at Jerusalem at the feast: for they also went unto the feast.
46So Jesus came again into Cana of Galilee, where he made the water wine. And there was a certain nobleman, whose son was sick at Capernaum. 47When he heard that Jesus was come out of Judaea into Galilee, he went unto him, and besought him that he would come down, and heal his son: for he was at the point of death. 48Then said Jesus unto him, Except ye see signs and wonders, ye will not believe. 49The nobleman saith unto him, Sir, come down ere my child die. 50Jesus saith unto him, Go thy way; thy son liveth. And the man believed the word that Jesus had spoken unto him, and he went his way. 51And as he was now going down, his servants met him, and told him, saying, Thy son liveth. 52Then inquired he of them the hour when he began to amend. And they said unto him, Yesterday at the seventh hour the fever left him. 53So the father knew that it was at the same hour, in the which Jesus said unto him, Thy son liveth: and himself believed, and his whole house. 54This is again the second miracle that Jesus did, when he was come out of Judaea into Galilee.
Considering 2 Kings 17, why did the woman at the well say that her father was Jacob?
Please support all answers with scripture not using sources outside of the bible for citations or reasoning as I am seeking that which comes directly from the bible.