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Write each ellipse in standard form.$$4 x^{2}+y^{2}-2 y=15$$...

Question

Write each ellipse in standard form.$$4 x^{2}+y^{2}-2 y=15$$

Write each ellipse in standard form. $$4 x^{2}+y^{2}-2 y=15$$



Answers

Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.

$16 x^{2}+25 y^{2}=400$

He's informed the constant this one. So divide Levin side by to get to two X squared over two plus y squared over two. It's one. Simplify it and get X square over one plus y squared over two. This one so a is one and b a squirt of two, which is larger than a so that sea will be. Be square, minus a score and then take the square root of it. So this would be what if we look at the plot? It will be like this. So here the verses are on the Y axis because B is darted and a and these two points are squared. 20 screwed up, too, and zero minus groat of two, The focus point. Also, it's metric, and these will be 01 and zero minus one.

So in this problem, we are asked to right this equation in that standard form and to grab the equation. So in the Senate for probably want a one on this side of the equation. So first thing that I'm going to do is I'm going to divide, but for on both sides of this equation, and so that is going to get me ex cleared over too first. Why squared word for Is he going to so two graf ISS, huh? This equation, It's going to be in a lips, so we need to find the values a MP. So between these two numbers, bigger one is going to be a squared. So in our keys, that's going to be four. And the other one is going to be b squared. That's too. Okay, So, um, the other number that we want iss A can be still taking square root of a squared. We have to. And Herbie, we have square root. I have to. So because the a iss under why that means that the focal access is going to be the y axis and the folk I air going to have organics, sirrah, and plus and minus a number. Uh, see where? See iss defined like this. So the next step that we're going to do here is find out who this seat iss. Um, so we just use this formula. We know that a squared ISS for and B squared ISS too. So this that means that sea is going to be squared up to So, um, let's get it graphing. So we said that Why accesses bifocal access and all right, here, we're going to find the focus. Um, so OK, are have co ordinates CEO and plus in Linus seat Bruce, Square root of two. So let's stay here. ISS zero comma, positive square return to and on the other side, we're going to have cereal and negatives Clarity, too. So that's our guy. No, uh, the Ellipse is going to go around here, but we want to know exactly where it crosses the my access that is going to be given by December. Right here. So, um, to see over here is to over here iss negative too. And on the x axis, excluding to cross over here, the value me. So let's see that we have It's, uh, probably closer over here It's negative. Square root of two. Never here is squared. So finally we have our ellipse. Its going to cross here on the value of B or negative B negative, eh? This should be closer. That iss, um b and again A Over here, something like this should be the graph for it. This equation.

In this problem, we're going to be analyzing the equation of the Ellipse that's given here. This equation is already written in standard form, and since we don't have X minus a quantity squared or why minus quantity squared, we know that the center for this ellipse is going to be located at the coordinates 00 Next, we're going to locate the vergis ease. We need to find the value for a and we use the fact that a squared is equal to 64 so a must be equal to eight. And we'll need next of value for B. We'll use the fact that here B squared is equal to 25. So the value for B is equal to five. So the quantity A equals eight corresponds to the major access of this problem. And that tells us we have vergis ease located at the following. We have zero positive eight and zero negative eight from the major access and from equals five, we get two more Vergis sees this will correspond to the minor access, and we have 50 and negative 50 for those for disease. Now we're a to determine the equation for the folk I This equation is going to be C squared equals a squared minus b squared. So in our case, we have c squared equals 64 minus 25. Or in other words, we know that C squared is equal to 39 altogether. Taking square roots, we have that c is equal to the square root of 39. And from this information, we now know that the coordinates of the folk I for this ellipse are located as follows. First note that the major access of this ellipse is on the Y axis from this portion of the original equation. And so our folk, I will have first cornets identically equal to zero. Then the second cornet will be squared 39 for the first point and negative square root of 39 for the second point. This completes the analysis for this ellipse

The first thing we want to do here is rewrite this in standard form. So I'm gonna rewrite this as X squared over instead of 25. I'm gonna rewrite that as five squared. If you need help trying to find with the perfect squares, you could always take the square root of each denominator. Our next faction is why squared over four squared, and this equals one. Now I noticed that the denominator of my X squared term is greater than the denominator of my wife square term. That means their exes are major access and our wise or minor access. So our endpoints we know that we're working with is we're going to have zero plus or minus a and A plus or minus B comma zero. So I am going to start plotting in my endpoints. I know that I have points at five, comma zero and negative five, comma zero and I also have points at zero comma four and zero. Common Negative for last up is to connect our points to draw our lips


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