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Zid the best line through the origin whih Ets the points (1,1),(2,1), (3,2), and (4,2).Suppcse the eigenvales of A are Jz'},$ and Does lin A" = 0? {Briefl...

Question

Zid the best line through the origin whih Ets the points (1,1),(2,1), (3,2), and (4,2).Suppcse the eigenvales of A are Jz'},$ and Does lin A" = 0? {Briefly justify your response:} Il_co

Zid the best line through the origin whih Ets the points (1,1),(2,1), (3,2), and (4,2). Suppcse the eigenvales of A are Jz'},$ and Does lin A" = 0? {Briefly justify your response:} Il_co



Answers

Let $L$ be the line through the two points $A$ and $B$ . Prove that $C=(x, y)$ is on the line $L$ if and only if $\overrightarrow{O C}=t \overline{O A}+(1-t) \overline{O B},$ where $t$ is a real number and $O$ is the origin.

We're going to do problem number three. And this question. What have to do? We have to just applaud the points. Okay, literally the points off. Question number three that we have to plot that point is zero comma zero. Another point is be that is one comma one on a third. One comma. One third point is C which you have to locate. That is trigonometry on. The point is management common man is one on e point is managed to come up. Managed to we have toe just, uh, the present. All these points on the graph on were to describe the set of all points of the phone. EKO my, where anywhere is a real Lombok. So I stopped. Will name the excess. This is why access a positive. It will be in Venezuela. Access. This is X axis a positive. It will be Manus X axis. Now, this is the center one, which is zero comma zero. Well, mark this as, uh, taking to unit as one. So this will be one. This will be to this will be three. This will be four. Same goes follow. So why exist? One to 345 Simple assault. Negative. XX is also manage one managed to manage three Manus for Anderson. Same follows. So man is way. Also, management managed to ministry and so on. Okay, so now let's start representing the 0.5 point zero comma. Zero means zero N Y zero. This is the center one. So write it as point A None other is be point, that is one comma one means access one. And why is one? So this will be the B point. Okay, Now, third point is three coma treatment success three and Y is three access three and weighs three. So this is the point where Access three, this is the point. Where who? I s three. Okay, so that's really the point on that point, we will representatives see now depended, uh, access management and wise management. So this one other point, this will be a point D Now you point is excess managed to and why is managed to So this will be another point. This point is e. Okay, So we had represented all these points, all these five points. Now we have to describe the describe the set of all points, uh, from a coma, a Okay, so this is, uh Oh, my all are these heads off a coma over A. Both X and Y are equal. Okay, A, B, C, D and E both have X and y are equal. Okay, Now we see one specific thing which we can saved from this all punishment and they both of equal in section way. Both are equal. Then let's see if we join all this points. Okay, So if you see this, then they are in the same line. Okay? The president today, same line. Okay, We need not to go through that. This is only for education. Just learning purpose now in a and A Okay. Both the distance from X x x and y axis are same. Okay? Distance from X x and y axis of the point are same. So we're done with discussion. Thank you very much.

All right. So you want to find two point on the line exit over a plus Y, Herbie? It's closest to the origin. What? Okay, so let's see here. If we saw for why we write this a za function. Why? Well, this is gonna be nineteen years being over a X plus being, and I guess we're assuming here that, um yeah, yeah, that's it. So why is equal to this? Which sea? We'LL just leave like that. Okay, so the distance from the origin we call that d as a function of X, it's going to be the square it of X squared plus y squared. But we have this expression for why squared so x squared us, they'd be over a x plus be squared, okay. And X, what's the domain of axe will Excuse me. Right. Because the domain of this line is all real numbers X. It's just any real number. Okay, so let's notice First of all that as X goes to plus or minus infinity, well, this is going to be behaving like a linear function. Stray but positive. So we're gonna have waken Basically fat factor out in X squared is gonna be an absolute value of X. So he's going to go to infinity. I mean, that makes sense Like his ex gets far away from zero. Of course, the distance is going to be far away from here. Okay, so let's take the derivative And hopefully we'LL just have one critical point and we'll know that has to be the minimum. Okay, so this is one over two square. It's X squared. What? Negative B over a x plus Be square, and then change will take the derivative of what's inside the square root. That's going to be to X plus two times negative. Be over, eh? Lisping. Okay. And if we want d prime to be zero the next there, they're going to be a very expert speaking. And then we have another change room seven times. Thank you. No. His ex costa plus or minus infinity goes to infinity. And in here we have another factor of so drew those two x plus bring the two down and then we gotta multiply by inside of this. It's going to be times the negative. Be over, eh? Okay. If we want d prime to be zero, we want this term to be zero because this isn't I'm going to be zero. So let's set two packs plus two times negative B over a X plus B times negative B a equal two zero. Okay, so this is two x plus two times b squared over a squared X plus minus two b squared over a Okay. And so let's see. We can divide through by two for sure. And then we'LL move this over So he has We can fact out an X v of X at times one plus b squared over a squared is equal to he squared over a And so I'm gonna write this factor as a squared plus b squared over a squared So the short one is a squared over a squared squared over Hey! And then So when we saw for X what we get me gets it. Okay, so we flipped this We can cancel in a so we have and being squared over Hey, squared plus b squared. Okay. And this will be the X value because the only critical point that minimizes the distance from the origin to that line and then the wind value to the y values you just plug it in. Intar equation here

Well, good morning there, or good afternoon, depending on. So let's go ahead and get started. We want to find a point and on the line X over a plus. Why, over be equal toe one, which is closest to the origin. Okay, so the first time I do is I'm gonna solve and put this in our why intercept form so wise, equal to b minus, be over a times X. So we know that that line is a decreasing line, and I'm gonna go ahead and graph in. So this point appear where he crosses thes e Y axes is zero B and down here, where he crosses the X axis is a comment zero. And so we know it is a decreasing line like that. So I do know that the point closest to the origin is going to be in this region right here, okay? And so we want to know that distance from the origin to that point and its closest, So that means the minimal distance. So I know that if I pick, um, just a reigned, uh, x y here on that line, and I'm trying to find that minimal distance, so I know that the distance is going to be this square root of X squared plus y squared. Okay. And so now we need to come up with that, um, either X in terms of why or why in terms of X. And we do know now that, um, why evaluated at some X value is going to be a B minus B over a times X. So this is going to be the square root of X squared plus B minus B over a X, and we're gonna square that. And then what I'm gonna go ahead and do is go ahead and distribute that, um, or multiply out two of the beam, honest the over eight times X and then combine like terms. And so when we do that, I will get X squared plus B squared minus to be squared over ace time fax. Um, plus B squared over a squared times x squared. And so now I'm going to collect my X squared. So my distance is, um X squared times one plus B squared over a squared. And then we have that minus to B squared over a times X plus B squared. Okay, so there have that. And so now we know that when we were looking for minimal or maximal, we're gonna have to take the derivative. And so my take the derivative of that distance with respect to X, and so that is going to be 1/2 times that x squared that entire square root. So we're gonna have minus that to be squared over a times X plus B squared to the negative 1/2 and then were multiply that by the derivative of what's inside. And so that's going to be to eggs times that one plus B squared over a squared minus to be squared over a. So we have that. Now we're gonna go ahead and simplify that. Simplify that a little bit. And so we have, um, that derivative with respect to eggs equal to X, because my twos cancel out right here. Um, Times one plus B squared over a squared minus, B squared over a divided by that square root. And so that's gonna be the X squared over one plus B squared over a squared minus to B squared over a plus B squared. Okay. And so now we want to know where that derivative equals zero. So that's primarily where the numerator is going to equal zero. So we're looking for that critical number. And so we have X times one plus B squared over a squared minus, B squared over a And now we are going to solve for X and we get X is equal to B squared over times a over a squared plus B squared. So there is my ex value. And now we're gonna solve for that. Why value? And so we have that. Why value got to get used to that new feature. Um, so why is equal to B minus B over a times that X value, which is B squared times a over X squared plus B squared? And so those A's cancel out, we're gonna put it over a common denominator. And so I have be a squared a minus or plus B cube minus B cubed over the A squared plus B squared. And so we have B times a a squared over a squared plus B squared. And so that minimal distance. The point, um, from origin with minimal distance is, um, the X value, which is a B squared over a squared plus B squared and then a squared times be over a squared plus B squared

We're going to do problem number four in this question. What they have to do, we have to just applaud the points. Okay? The points are given, that is zero comma. Zero be point is given. That is one common management. See, point is given that point is three comma ministry. The point is given that is manners One comma, one you point is given that this ministry comma three. Okay, so we have to describe all these sorts off ones which are having a element, etc. Okay, where is the real number of this form? Okay, so we will just blow the Gaff Brought the points first on the ground. So let us name the exist. This is why I exist. This is, uh, Venezuela exist. This is X X system. This is Manus X axis. The central on is zero. We will take to intercept one so you'll write her one. This is two this is three This is four. This is five same follow So negative xx is also okay. So we will write that also this is management. This is managed toe. This is ministry. This is madness for same follow. So wakes is also 1234 Same for negative. I also when this one managed to and ministry. Now let's start boarding the points question zero comma zero in sections. You don't Y zero. This is point a now point to be There is one common menace. One in Texas one anyway is medicine. So this will be another point. Let's say this is B See point is excess tree And why is ministry So this is the point where Access three and why is ministry is this one. So the point will be this one. So we'll point it is see nowadays access management And why is one so this this point d He is a ministry access ministry on what is three. So this is three. So this will be the point, which is B. Okay, So if we talk about these shirts off a common management, so all are in the form the forms are Let's see, this is that is, uh, this one that is B see on BNC only are the source of this where y is having a negative sign on access. Having this. Okay, If you look at like that, only now if you see this all cases also. Okay, Okay. It will be a negative number. Let's say it will be many. Say, then, the This will be manners off, man, I say. Okay, so it is holding this property the third. That is a common management. So what will? What we can say is we can say that all the number, all the number A, B, C, D and E will hold the set. That is a comma management. A common menace. Sorry. Okay, Now, if we join all this line, then they're going to make a straight line equation of a straight line. Okay, for a straight line. So this is the properties off these old points. I management a man necessary. Yeah. So this is the required solution off this question, That's all. Thank you very much.


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