consider-the-hyperbola-with-equation-fracy2a2-fracx2b21-a-show-that-ypm-fracab-x-sqrt1fracbx2-b-expl

Question

Consider the hyperbola with equation
$$
\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1
$$
(A) Show that $y=\pm \frac{a}{b} x \sqrt{1+\frac{b}{x^{2}}}$
(B) Explain why the hyperbola approaches the lines $y=\pm \frac{a}{b} x$ as $|x|$ becomes larger.
(C) Does the hyperbola approach its asymptotes from above or below? Explain.

Rakesh Kumar Sharma · Accepted Answer

in this worship, they push behind the will of human voices. What&#x27;s whereby, B squared. Minus excess. Where? Right? A swell you soon. Okay, so we have to grow some residents. Okay to the part airs. Actually, Felix friend, overvalued by special waste Where by this fiery actually won by excess for ways for cannot take as seriously on the route. Almost takes away people&#x27;s with post minus B who go one less excess. Where? By these American. Okay, well, except if I love it. So let&#x27;s take X by a common from this under. So we&#x27;re beginning. Why? Because of plus minus B y a X move over. 01 Yes. Exits friendly a square my excess for okay in the question. But the bees, you know ESPN is units. Be saying that because if you take that for you So you really busy and usually be ever be your ex Moved off in order to when this, uh, we squired by excess Where? So I ended up resentment. We have proved that is a very supposed to do in case today in part two is in Chile. Oh, yes. Saying the same that lord off expressed extra tend to use this experience to infinity, then be by means forever exists for reading groups. Maybe zero. OK, because actually be used. Most of this will be useful medicine refining, financial and every member. So the question will be converting toe. Am I being booed? More minutes. Well, simply, Why would be closer? Less minus a by B? Okay. Yeah, GTO. The human equation with this is and this is we&#x27;ll get the letters work. So there&#x27;s already a group of our P and Potsie in the same That that s a short because I died from the real world. Up upside. Get so should you need to make if you make that. Do you? So this is X. This is okay. Retired. You see this as some thought we touch from your in from war, takes all of the systems rainless. He maybe it&#x27;s OK for this s and your drink. That&#x27;s for boats and this form of all time as well. So that super tonight will be approximately intricate on question direction of expenses, that acceptance of their destination for special

Rakesh Kumar Sharma · Answer

in this question. You&#x27;re the hyperbole up there, this excess acquired by a splatter minus y squared by the sweat. It was a lot of things about India. So let&#x27;s try. The words of y square will be by the square will be actually with people school ex described by his fire minus one. Okay, And if it&#x27;s and if I am trying to get the value of the three day Andrewjkahn both side and we&#x27;ll be hope the situation will be blessed. Be excess five by a squared minus one. Okay, what you&#x27;re doing is we&#x27;ve been taking expert es communism. So we&#x27;ll be Get me. Why would you be opposed to bless minus be by a more off X in Doda one minus is far in excess for so that is a reserve. You are calling on the pushing okay, in part B on the question of the same. And if more off experts are ex, actually a few for injury there ness that is more of excess standing toe infinity. In that case, this accretion will be correctly when more defects, this more effects will be the world or actually being used member essentially this quantity so That is where, but excess? Well, this is huge. Why? I think this is something I need upon whose ordered a So this should be approximately zero. Okay, this one will be approximately zero. For In that case, we are getting writing because you place might be a Indo X. Okay? And that is the creation off ignitions off simple as indoors off this type of looking. So that is the second question we have to get in the southern part of the same. But essentially, the hyper hula will be achieved. So I would say OK, so if you try to find the half because sketch on the I grew up. So I did what I Maybe something like this, right? I will be something in this. Okay, do something like this. And if this is that something, that s something we actually catching on both side of the hyper. Do you see this two point? Then I think we should go soon. Be my A election. This I simple. We&#x27;ve been pitching. I wanna on move the sides off line along the left intake and basically is what we have been in question. So that is a company solution. All special

Ankit Gupta · Answer

here we are given with too hyperbolic equations, which are why is get over 16 minus ex scared over nine physical Diovan and why scared over 16 minus X scared over nine is equal to minus one. We were asked whether these equations are hyperbolic in nature or not. So the comparing the given equations with general equation, which is by minus at scared upon a scare minus X minus case care upon. Be scared is equal to one for the first equation. The equation fulfills E general equation we&#x27;re at is equal to zero, and K is equal to zero is equal to four and basic or to three now shifting duty for a second question we are given with by scared by 16 minus X care by nine is equal to minus one. Multiplying the equation with minus one. We get ex scared by nine minus off. Why scare over 16 is equal to one. The question is similar to the general equation X minus zero scare upon Be scared minus via I minus zero. Scare upon for scare is equal to one here. The transfers have seizes along by X is here it is equal to four B is according three and H and K is equal to zero. So both these equations are hyperbolic in nature.

Bobby Barnes · Answer

So in this problem they talk about how the Assam totes distance from the line. Very, and what they want to show is that it approaches zero. So they tell us to show that Miss Limit on the board approaches zero. So let&#x27;s go ahead and do that. So any time you have something where it&#x27;s like something minus the square root or screw minus something Um, the way to start is to multiply by the contra git of this so topping bottom. And in doing that, we would end up with X squared minus X squared, minus a squared all over. And then we shall have our conjugal ended denominator. So the X plus X squared minus a square square rooted. Now, notice how the X rays there comes out, and then the negatives air will cancel, so we&#x27;ll have be over a limit as X approaches. Infinity of base, where over X plus expert, minus a squared, all square rooted. Now a square is a constant, so that doesn&#x27;t matter it all. So all we need to think about is well, what happens in the denominator? Well, X squared goes to infinity. If we subtract a constant from it still goes towards infinity square rooting that would still go to infinity and then X also goes to infinity. So we have infinity plus infinity. So this should really be over a a squared something going to infinity. And well, in this case, that would just be zero. So you showed what they wanted. So we keep our little proof box and smiling face because we&#x27;re glad that we&#x27;re done.

Bobby Barnes · Answer

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&#x27;t pull off the B squared is because if we don&#x27;t know if he is possible negative and we would have to have this absolute value and I just don&#x27;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&#x27;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&#x27;m gonna claim that e positive square root of positive line and the negative square root cause with the negative life. So let&#x27;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&#x27;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&#x27;ll be plus, as opposed to mites and they remember any time multiply. We also have to divide by the conflict because we can&#x27;t multiply by anything other than one. Okay, so we end up with that&#x27;s here now when we go ahead and multiply out, remember, that&#x27;s just the difference Squares. So we&#x27;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&#x27;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&#x27;m going to say be a infinity like that. And the reason why I&#x27;m saying be over a is well, if be divided by its negative, we&#x27;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&#x27;re going to do instead, it&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;re going to be left with zero over eat over a plus. He&#x27;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.

Christopher Stanley · Answer

now on this problem. We want to identify the aspirin totes of this hyperbole. Now, for me, if I&#x27;m gonna identify the accent Oates, I want to graft this. Yes, let&#x27;s do that. Notice your center of 00 That&#x27;s the export in the white corner, the making both zero underneath the exes of four. But since it&#x27;s inside that princes, it would really be a 16. But we care about the force or the report of the left and four to the right. Same idea with their with a why it&#x27;s why over five squared. And so we go up and down five. And so this would give us our box only reason to care about the boxes because we care about its diagonals. That should go through that point right there. So drawing on that other Daniel, I said I should go through those corners there now. I don&#x27;t even, you know, graph the rest of it because all I care about here are these Assam totes notice they both throw through that point there in the center. So if we don&#x27;t find the equation while I need no point on a slope, your point So we&#x27;ve got to ask, intones, Here&#x27;s 1212 The point. They both go through a 00 and then to find their slow things. Sofas rise over run. So it&#x27;s going up here. And then over there we went up. Five. We went over four, and so our slope on his 1st 1 this 5/4. So if you find the equation of the line and soap intercept form through 00 with slope of 5/4 B y equals five force. Yes, now here with the other one. You went down five and right for So it&#x27;s the same idea. Your soap is just negative. 5/4. So our equation line here with the white was negative. Five Force. That&#x27;s when you find that equation line through 00 with a slope of negative 5/4. And so those would be the two equations over at Simpson&#x27;s

Problem

Let $F$ and $F^{\prime}$ be two points in the pla…

Problem 50 Hard Difficulty

Answer

Related Courses

Precalculus: Graphs and Models

Related Topics

Discussion

Top Algebra Educators

Catherine R.

Kayleah T.

Caleb E.

Maria G.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 11

Video Transcript

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Precalculus: Graphs and Models

Catherine R.

Kayleah T.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Precalculus: Graphs and Models

Catherine R.

Kayleah T.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

Raymond A. Barnett, Michael R. Ziegler, Karl E. ByleenPrecalculus: Graphs and Models

Catherine R.

Kayleah T.

Caleb E.

Maria G.

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Precalculus: Graphs and Models