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University of North Texas

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Problem 73 Hard Difficulty

Show that the lines $ y = (b/a)x $ and $ y = -(b/a)x $ are slant asymptotes of the hyperbola $ (x^2/a^2) - (y^2/b^2) = 1 $.

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Video Transcript

we want to show that the lines. Why is it reminded? Be over eight tones X Are these like acid tubs of hyper Bhola squared over a squared minus y squared over peace. Where is he? So what I would have had and did to start waas rewrite the equation just in terms of why which we end it with this square root with the plus or minus. So the reason why I didn't pull off the B squared is because if we don't know if he is possible negative and we would have to have this absolute value and I just don't have toe right absolute value every now and then. So, uh, just kind of keep that in mind If you do pull it out. When you're writing this, there needs to be absolute value of being on the outside. Uh, but if we want to show that those are the slight ass because what we need to show is that this function here, minus one of those lines is equal to zero when we take a limit SX a purchase infinity. So Ah, and I'm gonna claim that e positive square root of positive line and the negative square root cause with the negative life. So let's go ahead and go through how the take these limits. So you might remember from the chapter when we talked about infinite limits, that the technique we should use for this is to multiply by conscience. So let's go ahead and do that, because by multiplying by the contra get we get rid of that square root. And remember, the congregate is just We take the middle term and make it the opposite side. So it'll be plus, as opposed to mites and they remember any time multiply. We also have to divide by the conflict because we can't multiply by anything other than one. Okay, so we end up with that's here now when we go ahead and multiply out, remember, that's just the difference Squares. So we're going to be left with. Well, the square roots will end up. Just cancel no, essentially And what be squared times X word over a squared minus one and then minus B A. But those were X squared and then all over our conscious yet still peace word X squared over a square mile swim square rooted plus B for her a X No notice. If we were to distribute this b squared, then we end up with a positive p X squared for a squared. So that ends up counseling out with that there and we're going to be left with the numerator. Just be square or a negative piece works a lot of negative b squared and then all over our conscious kit again. B squared times, expert of his bird, minus one square rooted must be over a X Now, If we were to take the derivative of this directly, you might notice that will end up with infinity. Plus, I'm going to say be a infinity like that. And the reason why I'm saying be over a is well, if be divided by its negative, we're gonna end up with unity. My symphony and I have no idea what that means. So just to kind of get around that what we're going to do instead, it's first multiply the top and bottom of this by one of her ex. Before we actually applied the limit. So doubt doing this, we'll have the limit as experts pity of negative e squared over X all over it. Remember when we pulled that one over X into the square root. We have to square it. So that's why it will cancel out with the X squared that we have in there. So be square will be won over a squared minus one over X word square root and then just plus B over. And I noticed when we take the limit of this well, the new mayor goes to zero. That there goes to zero and we're going to be left with zero over the square root of B squared over a squared plus B over a, which is zero so that their shows that be over a X is one of these slant pass symptoms. Now, for this next one here, we can go ahead and follow the same steps. So first noticed that these two negatives there counts and we can go ahead and rewrite it with the B A or B, divided by a X in front and in follow the same steps. So just not to kind of boreal with algebra again. I went ahead and did this beforehand. So in this first up here, just like we did before, we multiplied by congregate and after that everything would simplify down to just be squared times one of her ex or be square over contra git. And what we're going to be forced to do once again is multiplied by one of her ex. And we end up with this equation here. And then again, we just go ahead and take the limit as excuse infinity. So the numerator goes to zero. This part of the square root goes to zero, and we're going to be left with zero over eat over a plus. He's where, over a squared swear rooted, which again is a cigarette so that their shows that the other line is also a slant acid, too.