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Sketch the ellipse, and label the foci, the vertices, and the ends of the minor axis.(a) $9(x-1)^{2}+16(y-3)^{2}=144$(b) $3(x+2)^{2}+4(y+1)^{2}=12$...

Question

Sketch the ellipse, and label the foci, the vertices, and the ends of the minor axis.(a) $9(x-1)^{2}+16(y-3)^{2}=144$(b) $3(x+2)^{2}+4(y+1)^{2}=12$

Sketch the ellipse, and label the foci, the vertices, and the ends of the minor axis. (a) $9(x-1)^{2}+16(y-3)^{2}=144$ (b) $3(x+2)^{2}+4(y+1)^{2}=12$



Answers

The ellipse $\left(x^{2} / 16\right)+\left(y^{2} / 9\right)=1$ is shifted 4 units to the right and 3 units up to generate the ellipse
$$ \frac{(x-4)^{2}}{16}+\frac{(y-3)^{2}}{9}=1 $$ a. Find the foci, vertices, and center of the new ellipse. b. Plot the new foci, vertices, and center, and sketch in the new ellipse.

Part a find the focus points. The Vergis is in the center. So the old lives a cz something like this where this is four and this is three and we are moving. It's four units to the right and three units up. So the old focus points were calculus. See iss Croteau, squirt of 16 minus night, which is screwed of seven. So this originally was screwed of 70 and now we move it to the new lips and got the new focus. Points are four plus growth of seven three, the other one for minus words of seven three and the birth asses. This was the Vertex before it was 40 So now the new purchases are 83 and 03 So simply what we do is at four to the first demon. And then at 3 to 2, the second element for everything the center dissenter was 00 before. So now the new center is for three, and then we draw the new lives like this. First you'll find the center, which is 43 and then find the two over taxes, which are 83 and 03 So then this become simple. This is the new your lives. This one should be in contact with the X axis. And this 0.83 this 0.3 the center is 43 I'm not gonna market here. Um, and we also need to mark the focus points which are for Post Screwed of seven three and the other one is for minor squirts of 73

In this problem, we're going to be fully analyzing the equation of a lips that we see here displayed. First thing we're gonna note is that the center of this ellipse is going to be at 00 The reason for this is that there is no extra terms surrounding X and why to the power to next. We're going to analyze the Vergis ease of this ellipse. And to do that, we need to determine which denominator we'd like to associate with a squared. I'll pick the denominator with the largest magnitude as a squared. So that's the opposite denominator. Nine will be denoted as b squared. It's no we have that. He squared is equal to 36. B squared is equal to nine, which implies that a equal six and B equals three upon taking square roots. Let's also note further that in this situation, the major axis is the X axis, since a squared is associated with the largest quantity for variable X. Now we're ready to determine what the folk I should be for this equation. To determine the folk I we used the equation that C squared equals a squared minus. B squared in our situation C squared will then be 36 minus nine or C squared equals 27. Upon taking square roots, we have that see is either equal to positive or negative squirt 27 which reduces to positive or negative three square root of three to the nearest 10th. This is approximately equal to positive or negative 5.2. What this information tells us is that the locations of the folk I are going to be the following. First, we have to determine whether the folk a we're going to be on the X axis or the Y axis. But since the major excess is the X axis, we can put in zero here and here. So the points of the folk. I lie on the X axis itself and they'll be at locations three. Route three and negative three Square root three. Now we're ready to begin plotting. First, let's put down the locations of the Vergis ease will use this information here where a s six giving us an X intercept here and ex intercept here at negative six B is three corresponding to the minor access. So we have a wide receptor at three and a y intercept debt. Negative three. Next we noticed noted that the Vergis ease air located at positive or negative three Route three, which is approximately equal to five square to to so the folk. I are going to be roughly here and here. Now we're ready to begin sketching the curve for this ellipse with the rough freehand sketch. It would look about like this going piece wise, and this is our full analysis for this ellipse.

We have problem number seven. We do skits, the lips and the level of folky vortices and ends off minor access. My Texas. Okay, so find a question number A is X squared by 16. Plus. Why square benign equal to one, which is in the form off access choir by a Squire plus y Squire by B squared equal to one. After comparing, these two will be getting a square equal to 16. Which means if we take only positive value so he will be equal to four. Or we can take negative value as well. No problem. So but we're taking positive value. Okay. Similarly, base quite equal to nine, which means we will be equal to three. So first of all we have our center center is zero comma zero. Of course. Okay, what? This is and and points off mine that exists. Okay, So what? This is our what is it will be plus minus a comma zero that is plus minus four comma zero, which means minus four comma zero and four comma. Zero. There were two burgesses or we can say and points of measure exists and points of minor exists for mine and myself. Minor access will be zero command plus minus B, which means zero command plus minus three. So zero comma, minus three and zero comma plus three or simply three in the documentary. Okay, Now we need to find the for Kai. So for this, we have to calculate we have to calculate C square equal to a square minus b square A square mints have been provided 16 minus nine, which is seven 16 minus nine, which is seven. So she will be equal toe plus minus seven. So we have four k equal toe plus minus C comma, zero. So plus minus until seven, comma zero. So this is the focus under road seven, which means approximately 2.6 we can say plus minus 2.64 six comma zero. This is the focus. Now let's draw this parable. This ellipse. Okay, this is our Y axis. This is our X axis. So this is minus one minus two minus three minus four. 1234 This. 123 for minus one minus two. Minus three. And of course, minus four will be somewhere here. Okay, so now we need to level the folk I the word he says so. First of all, let's level Vertex what it is, which are plus minus four comma zero. So this is what X number one, let us say vortex number one minus four, comma zero. This is what X number two. How much This is What X number two that is. We two for common zero and finds a major Texas my nexus. Uh, zero command plus minus three. So one is zero comma tree Another is this zero comma minus three? Okay, for Kai, this is the focus. Uh, plus minus 2.646 So somewhere hair 2.6, this is F one. This have to a fun is two point 646 with negative sign comma zero and after is 2.646 with positive sign. Okay, so let us draw a line four lines to make a rectangle and the line should pass through all the four points which are endpoints of matrixes and then points of my Alexis Or Okay, so now the A lip should just touch these lines over here at four points. So it will be like this. Okay. Perfect. Until now. Perfect Okay, this is our lips. Now comes part B. Part B is nine x square plus y square equal to nine. That is, divide both sides by nine. We're locating access Square plus y squared by nine equal to one. So we'll be comparing with with access choir by B squared plus y Squire by a square equal to one. So after this. Okay, so in this case, since the denominator of wire square is greater than denominator of x square. So our lips will be along y axis I'm in major exes will be along y axis. Okay, so in this case, be a square is one. So be is one ISS square is nine. So it will be equal to nine or 33 three. So first of all, we need to find the folk. I Okay, then what he says then and some metrics is So, uh, we just find out C squared equal toe a square minus b square. That is nine minus one, equal to eight. So she will be plus minus on about eight. Okay, just take positive value. Do not to get confused to root two, which means two into one for two points to eat. 2.2 it almost approximately. I am talking about approximate value. So let us start with folky for car will be plus minus zero. Commander of 60 Commander of C zero comma plus minus C simply so. Zero comma plus minus 2.28 After that, the vortices What is this? Will be the end. Points off, major exists. Zero comma plus minus A that is zero comma plus minus three. Okay. And points of minor access for endpoints. Off minor access. It will be plus minus B comma zero. That is plus minus one comma zero. So these are all the points we need. Toe show on the graph. Okay, this is our Y axis. This is our X X is zero minus one minus two minus three minus four 123 minus one minus. Tool minus three. What? Not Minus is a plus. This a possible access. Okay. Minus one minus two and minus three. Okay. No, let us right. First vortices zero comma plus minus three. So one of the vertex will be here. This is zero comma tree. Another will be here. Zero comma minus t. Okay. Um, and points of mine that exists. Plus minus one comma. Zero off. One will be here minus one comma zero and there will be here. One comma zero. Now fork. I will be at zero gun plus minus two point to it. So somewhere here will be the focus. Zero comma, 2.28 and zero comma minus 2.28 or too rude to one. And the same thing. No. Let us make a box by using the points mhm which are in points of major axes and points of minor access. Like this. Okay, so a box is completed. So this is the reference line for drawing our lips so Ellipse will be okay. Is so this is the lips. Thank you so much.

Several Number eight question number is access squired by 25 plus y Squire, before equal to one. We need to escape this. Okay, so if we just compare this with access Choir Bye a square plus y squared by B squared equal to one, which is general equation. So a square will be equal to 25 so it will be equal to five. Similarly, the square is equal to four, so be is equal to two. Now, let us find, uh, required thing which are being which are to be find out. Okay, so level for Kai. So, first of all, see, square is equal to a square minus the square. That is 25 minus four, which is 21 soc will be equal to under root off 21. So under root off. 21. Well, approximately equal to on the road off. 21. That is 4.58 4.58 Now let us right. All the things which what we found. Okay, okay. Plus minus. C comma zero. That is plus minus 4.58 comma zero. This is the focus, too Focused. Okay, uh, then vortices What checks will be. What is It Will be a long X success. So plus minus a comma zero that is plus minus five comma zero. Yeah. Okay. Uh, now ends off. Made it exists. And so mhm measure exists. Will be zero comma plus minus B that is zero comma plus minus two. These are the ends off. Measurex is Okay, So now let us draw. We have five comma zero. Okay. So let us. This is X success. This is why access 12345 This is minus one minus two minus three minus four minus five. This is minus one minus two minus three, 12 and three. So first of all, focus that it's okay on 4.58 common zero. So it will be somewhere here, 4.5. It will be somewhere here on with Negative sign will be somewhere here. So this is our F one. Focus number one that is negative or 4.58 Common. Zero focused. Number two, negative or positive off. 4.58 comma Zero. This is These are minus five and five at end points off. Or we can say what this is. So even is equal to minus five comma zero and a two is equal toe five common zero and points of minor access. The zero command plus minus two. So B is between equal toe zero comma to and this would be one with equal to zero comma minus two. So let us draw the lines like this, which which will be perpendicular to the excess and passing through and points off. Major and minor exists to make a box. Okay, like this. So box is completed, so our ellipse will look like this. Okay. Okay, So this is our ellipse now comes. Pardon me. Part way is for X squared plus y squared equals statistics. So after dividing both sides by statistics, we will be having four x esquire by statistics plus y squared by 36 equal to zero equaled one which means access choir by nine plus y square by six. Why scribe statistics equal to one which in turn will be equal toe access square by three square plus y squared by six square equal to one. Now let's compare this with excess square by be square plus why square by square equal to one, we will be getting equal to six and be equal to three. Okay, so she will be equal to under route a square. Mines be square. That is, uh, 36 minus nine, which is under 27. Yeah. Okay, so under 27 on a three and all three are on the route off 27. 5.19 are approximately five point to approximately 5.2. So out for Kai, one more thing to be noted. That and the denominator of access choir is less than denominator of wire square. So the vertex will be along. Why exists? Okay, folk. I will, with zero plus minus, see zero comma plus minus 5.2 or placement three or three. But on the same thing. What this is will be zero comma plus minus A that is zero comma plus minus six and end points off. Minor access. Minor access will be 00 sorry. Plus minus B comma. Zero plus minus. B common zero. Okay, this was also mine. Rex is not major exist and points off. Minor axis. Okay, so this is a three. Okay. To plus minus two comma zero. Now let us draw their lips. Six comma zero minus six. Common zero. So and all along the Y axis. So let us is young. This is X and this is why this is 123456 minus one minus two minus three minus four, minus five minus six one toe minus one minus two minus three 12 and three. Now let us did not. The Vertex vert says zero command plus minus six. So what will be let us say this is a one zero comma. Six a to zero comma minus six. Be one will be somewhere here. The one will be two comma zero and be too will be that the endpoints of manner exists minus two comma zero and our folk I will be zero plus minus 5.2. So let us say this is F one zero, comma 5.2 and the local my minus 5.2. This is F 20 comma minus 5.2. Now let us draw a line. Okay, which will pass through all the four corners off measuring minor axis to make it the box to make it like a box which through which it will be convenient to draw. I was ellipse. Okay, so our lips must look like Okay, this is upper half part. Yes, Thank you. So


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