synthesis-show-that-a-hyperbola-does-not-intersect-its-asymptotes-that-is-solve-the-system-of-equati

Question

Synthesis
Show that a hyperbola does not intersect its asymptotes. That is, solve the system of equations
$$\begin{aligned}
&\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\
&y=\frac{b}{a} x \quad\left(	ext { or } y=-\frac{b}{a} xight)
\end{aligned}$$

Ankit Gupta · Accepted 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

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

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.

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

Ankit Gupta · Answer

Why is glad minus X plus one squared divided by before it was too well. This is a Christian, all hyper bola. We write in the standard form. So why minus zero? Fully square divided by one s quest minus X plus one squared, divided right food. It was +212 years age because no one and equal here too. And sequel to hear one plus four That thing Fruit and the fight. So So, yes. Then the seaQuest escape. So here, that is at Kay here. This vertical graph. This is where people were digging. So 10,000 people to hear minus one minus one. And deedle center on work takes vehicles here that is at on K plus minus a. So working because toe me when you go from here, that is really close to minus one on geo minus one minus one and minus one. That is minus one on me too close to here. Minus one minus one. And Europe last one. That is one overtakes and Folke was here poky because to hear that is formulate X on G plus minus c. So you, with this formula for four key So when it comes to here minus one on the minus one. Minus minus. Root would fight. So this is minus five on a frequent here, minus one on route fight. This is projecting, and I am. Yeah. Why minus? Do you go from here? Last minus. And here be a baby toe. Eight year one. And because too. So this becomes their half multiplied here and X plus one. So why was, too? Why close to X by two plus on a why he was here minus X way too X ray to a minus if we take me getting intent on and the quantity. So this is our and we have to draw the draft. So we take the point. You see, Jack sexy. This is excess. This is why exit and why does This is origin. We have work takes in here. That is work acceptable to minus one minus one. And he had this easier toe. Take point. This is one. This is two. This is minus one. This is minus 21 toe and this is minus one and minus two to the point, minus one and minus one. So this is the point. Yes. This is minus one and minus one point on minus one and XXIX minus one. And why? Because one and why Texas, This is the one. So this is the point. Minus one and one. So the graph for the Grafite vertical here to graph passes two of these points on in downward. You also passing for this point. This is the drop off hyper bowl off. Given a reason why a square minus explorers one foley square divided were fooled You 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

Synthesis Numerical Relationship. Find two number…

Problem 95 Hard Difficulty

Synthesis
Show that a hyperbola does not intersect its asymptotes. That is, solve the system of equations
$$\begin{aligned}
&\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\\
&y=\frac{b}{a} x \quad\left(\text { or } y=-\frac{b}{a} x\right)
\end{aligned}$$

Answer

There is no number $x$ such that $\frac{x^{2}}{a^{2}}-\frac{\left(\frac{b}{a} x\right)^{2}}{b^{2}}=1$,

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There is no number $x$ such that $\frac{x^{2}}{a^{2}}-\frac{\left(\frac{b}{a} x\right)^{2}}{b^{2}}=1$,

Algebra and Trigonometry

Lily A.

Kayleah T.

Joseph L.

Michael D.

Continuity - Overview

Continuity - Example 1

Judith A. Beecher, Judith A. Penna, Marvin L. BittingerAlgebra and Trigonometry

Lily A.

Kayleah T.

Joseph L.

Michael D.

Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

Algebra and Trigonometry