a-prove-that-the-hyperbola-fracy2a2-fracx2b21-never-intersects-its-asymptotes-b-i-fracy216-fracx294-

Question

a. Prove that the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ never intersects its asymptotes.
b. I $\frac{y^{2}}{16}-\frac{x^{2}}{9}=4$ a hyperbola? Is $\frac{y^{2}}{16}-\frac{x^{2}}{9}=-1$ a hyperbola? Explain.

Ankit Gupta · Accepted 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

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.

Ankit Gupta · Answer

in this caution were given with hyper bulla excess, spurred by a square minus y squared, divided by B squared equals 21 and it&#x27;s a simple given by a bike was to be divided by a X notice, considered it as a question of move on and it as a question about two. So this substitute the question off a sim toward that is why it was Toby by a X in a questionable one. And we get X squared, divided by a squared miners, be by a X. The Holy Squared, divided by B squared, equals to one these employees X squared, divided by a square. Miners be square X squared, divided by a squared B squared equals to one. These employees excess squared, divided by a squared minus X squared or debated about a squared equals to one these employees zero equals to one, but zero can never be equal to one. That is not possible. And when we come across such condition, these tells us that there is no solution for this, and then we don&#x27;t have any solution for a system off linear nonlinear equations. That means a seam thought and the hyper bola again neighbor intersect and spoon

Sriram Soundarrajan · Answer

Hello. You are never going to solve our problem. Number 15 here doesn&#x27;t bring us in our call. This hyper Bhola is a Cuban, but hi purple eyes gonna bake, of course, but he s square minus my square by square called Cooper for standard the you&#x27;re a square A could be throw standard that will be going through Yes. Oh, my b by year in the U. S. Here, plus or minus, year by year in tow. Learn if one day doing at the very cold wave at the other end while you go to minus six year, there&#x27;s no I&#x27;m gonna go to one. I am Greek going through mine, isn&#x27;t you? Both are happening that direct all their sloping big toe minus like weren&#x27;t indoor My toughened. You do minus our Yes, there are particular. Thank you.

Ankit Gupta · Answer

X plus one Holy Squared divided by nine minus. Why I minus the square divided by 16 If you questo one is a Christian off Khyber Bola on center sequence here at Cape and it is here, that is this is here minus one and minus three goes to zero. So this is This is minus 13 and a question here toe. And because here, food on a sequel to Root and, uh, yes, plus B s quest. So sequence here fruit and that 1916 25 Because toe fight. So sequence here fight and equals toe do. And we got to here. So here, Santa this year. And we have to find also Varteks. So working. No, that&#x27;s where next week was to that These Yeah, last minus is and you hear So this is working, so we want to hear that. You hear? That is minus one and eight years to hear. So blessed minus two. So this we come here. Where? Texas going to minus full on this we come here, that is minus one and minus three. We take here that is at stake here. So the left hand and right hand. So this becomes minus four and hitting and retweet questo here. Two and three, two and three on folky goes from here he goes to hear that is looking at plus minus C and this is case. So he was a 20 question here. That happened because two minus one a minus five year. So minus six and three on two. Did you want to hear four and three? This is Foky and Jim. Why you got to last minus B by a 20 Plight X. So here. Why, my nuts? You got to plus minus and be here for and two. So this becomes here. This is their equal here. Xa minus one minus one and PD. So centering. Here. Minus one. Entry on. Be here because food and equal to two. So here. Why? Because two plus minus B A upon X here for vehicles, toe food and take 1 to 2. So on this becomes here. Cool. You waited way equals here. Three equals here. Three So food divided with three multiplied X plus one. So this becomes here. If you take one time negative and positive, then why was too fast? We get with dignity. Gated. So why becomes here. What? X by three. Last 583 And why? Because to hear if you take Yeah. Would you keep saying then would x by three and plus hurting by three. This is our agent. Oh, we have to draw the graph. The X axis, the excess on. This is why on this in here, why does so the point of death were Texas here minus 43 This is minus full and three and 23 So worth it. Take decision one. This is two. This is three. This is full fight. 123 full. And this is minus one minus two. Minus three minus. Food minus one minus two minus three. Minus four minus. Fight to appointed. Here. Vertex minus four and three. This is minus full on. Why? Axes? This is three. So the point of this is here. This is minus four and three point and another point here, that is two n three xx is to And why exes? Because three. So this is the point here. This is the 0.0.2 and three point. So the draft hyperbole that is this passes through these points here. So this is that draft up Hyperbole off, Giving you grace and experts one divided by nine minus X plus one. Bullet squared, divided by nine minus Y minus three. Will you wear the right away? 16 Because one this is a graph off this given equation.

Suman Saurav Thakur · Answer

they said, Remember, 62 we have you in the hyper bola excess squad by four minus y squared. But it be a spotty called one is given Fine, Be so that seems towards our perpendicular to each other So isn&#x27;t or should have been x equal to minus me by X and B by X. In this case, a symptoms will be Why call toe minus B by two X. I call Toby by two weeks. So these are the slopes off the same towers for this has imposed to be perpendicular minus B by two in tow, be by two. Should be minus one because multiplication of store slope is minus one for the life too perpendicular. So this will cancel out be a square will be equal toe for so people were equal to two. Okay, so who is the answer? Well, thank you

Problem

Air Traffic Control Suppose you are an air traffi…

Problem 46 Medium Difficulty

a. Prove that the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ never intersects its asymptotes.
b. I $\frac{y^{2}}{16}-\frac{x^{2}}{9}=4$ a hyperbola? Is $\frac{y^{2}}{16}-\frac{x^{2}}{9}=-1$ a hyperbola? Explain.

Answer

Yes
No

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