Why did Tyrion Lannister go to his father's bedroom at the end of Game of Thrones season 4? Did he go there just to kill his father? Or was it after seeing Shae that he decided to kill him?
Tyrion Lannister needed answers. That was his nature.
Tyrion Lannister is a highly intelligent, thinking man. After the agony of being victimized by his nephew, accused by his sister, and arrested by his father, it became pretty obvious that Lannister internal family politics were in deadly play. This played out in slow motion agony with the trial that had one liar after another come up and testify against him. Tyrion knew that his father had a lot to do with this. Additionally, he was particularly victimized by Shae when she stood up and lied. I am sure he wondered about that.
When he was free to secretly move about, it was perfectly natural for him to seek answers. He knew where his father could be, so he went there. He needed answers.
In my opinion, he did not necessarily expect to find Shae there. But the betrayal could go only so far before he sought blood. It started with her. He then noticed the crossbow mounted on the wall. That was how he was going to “talk” to his father -- armed. The humiliation of a public farce of a trial -– and taking Shae –- was the last straw. His answers might well come in the form of an arrow tip.
In the book where this scene took place, he was particularly sensitive to his father's use of the word whore. After Tyrion asked where his first love (not Shae) was sent, his father said something to the effect of “Where do all whores go”. That prompted the patricidal shot. In this world, having the woman Tyrion once trusted there, combined with his father's attitude, was a very bad thing for the father’s health.
In the books, Tyrion went to meet his father to get answer regarding his first wife, Tysha.
“So just tell me something, and I’ll be on my way. One simple question, you owe me that much.”
“I owe you nothing.”
“You’ve given me less than that, all my life, but you’ll give me this. What did you do with Tysha?”
He does not even remember her name. “The girl I married.”
“Oh, yes. Your first whore.”
Tyrion took aim at his father’s chest. “The next time you say that word, I’ll kill you.”
“You do not have the courage.”
“Shall we find out? It’s a short word, and it seems to come so easily to your lips.” Tyrion gestured impatiently with the bow. “Tysha. What did you do with her, after my little lesson?”
“I don’t recall.”
“Try harder. Did you have her killed?”
His father pursed his lips. “There was no reason for that, she’d learned her place . . . and had been well paid for her day’s work, I seem to recall. I suppose the steward sent her on her way. I never thought to inquire.”
“On her way where?”
“Wherever whores go.”
Tyrion’s finger clenched. The crossbow whanged just as Lord Tywin started to rise. The bolt slammed into him above the groin and he sat back down with a grunt.
A Storm of Swords, Chapter 77, Tyrion
I think Tyrion was interested in the answer but Tywin's answer made Tyrion very angry and he killed Tywin.
I believe that he was already seeking out his father, and Shae was unexpectedly there, so she wasn't the determining factor, though that certainly escalated things.
Remember, his entire life view of relationships was cynically formed when the one he thought was his first love was, he thought, a participant in a ruse, as a whore, and didn't really love him. When Jamie visited him in his cell, Jamie finally told the truth that the encounter wasn't staged, and that Tysha was not a paid whore, purchased for him to lose his virginity. And, of course, that also made what was done to his first love afterwards a massive gang-rape and humiliation of an innocent peasant girl, not a whore earning money by doing her job.
He went back to confront his father about this. As posted by others, he specifically wanted to know what became of her afterwards, as well. Certainly seeing Shae betray him, twice (once at his trial, which he assumed was merely because she was afraid for her life, and then with his father), and finding out the truth about Tysha challenged all of his views about life and relationships.
I don't know if he sought information hoping to try and contact her, a Westros-FB friend request, or if not knowing was going to eat at him. I don't know if he even had a specific goal in mind for the confrontation, other than to tell his father that he knew, and what he thought of him, but it was Jamie's revelation about Tysha that caused Tyrion to not just flee.
Haven't gotten that far in the books, but as presented in the show the scene makes no sense. If he went to his father, who contrived the entire trial against him, to get answers, he wasn't going to get any. He was unarmed until he got to the room. His father simply would have had him thrown back in his cell, if not killed him on the spot. This was Tyrion's only shot, and a long shot at that, to escape execution. The only thing that would make sense is for him to get out undetected. Out of the entire series, this one scene stands out as the most contrived confrontation. The show mostly does a great job of avoiding this type of story-telling mistake.