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Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions.$$x^{2}+y^{2}=16$$...

Question

Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions.$$x^{2}+y^{2}=16$$

Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$x^{2}+y^{2}=16$$



Answers

Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$x^{2}+y^{2}=16$$

All right, we got these. Two equations were supposed Teoh graft them both and then find their points of intersection improved that those points of intersection satisfy the equation. So when he used us most that time here eso our first equation waas x squared close y squared it will 16. And our second equation is X minus y equals four. An easy way to graph. Um, we could see our points of intersection are zero negative. Four and 400 negative. 440 All right. And then we have toe prove that they both work here. So, um, let's do with zero Negative Force zero squared would be zero negative. Four square would be 16. And that does equal 16 0 plus 16. Let's try the 2nd 1 Foursquare is 16 and zero squared is zero, and those do add up to 16 over here. With the second equation, we've got zero minus negative four, which is like zero plus four, which is four. And then we have four minus zero, which that's also equal before. So all those points work in both equations

Equation that doesn't start out looking like the standard form of any lips. But we're gonna make it look like that standard form that I have appear. The ah X squared, divided by a squared plus life squared divided by B squared equals one. The equation for this problem were asked to Graff is X squared plus four y squared equals 16. In order to make it look like the standard form that set equal toe one. We're gonna start by getting in one on the right side. And to do that, we're gonna divide by 16. And remember, our equation property is to divide everything by 16. So I have a one now on the right side with the X squared divided by 16 that is just going to stay X squared, divided by 16 plus sign four y squared, divided by 16 4 divided by 16 would get me a four in the denominator and a Y squared on top. Now we have our standard form of any lips equation. We can use the A squared and the B squared values to help get our intercepts. So the number under the X squared 16 will call a squared and plus or minus four would give us our A value or our ex intercepts. The four is our B squared value, which would make B plus or minus two. These are our why intercepts? Because they're under the Wye peace in my standard form, with our standard form of our lips equation with just the X squared and just the wise squared on top that would make this ellipse B centered at the origin. 00 Let's go to a grid scroll over here and sketch this craft out So I have my 00 CenterPoint. I'll put my intercepts. I'll do the four negative for no particular order here. Two and negative too. So I have my ex intercepts my Y intercepts and I'm gonna sketch. This ellipse will do the top half the bottom half and here's my lips centered at the origin. So this graph of this ellipse is the equation X squared plus four y squared equals 16. Or, as we saw our standard form equation X squared over 16 plus y squared over four equals one

Equation for us, and this equation is gonna turn out to be an ellipse. So have the standard form of my lips equation appear to remind us that it should look like X squared over a squared plus y squared over B squared equals one. The problem that we're looking at is X squared plus four. Why squared equals 16. This particular equation is not in the standard form of my lips. So what I'll have to do is get it to look like that. And I'm gonna have to divide by 16 to get a one on the right side. But my equation Properties also tells us to divide 16 on the left side as well to both of those pieces. So my X squared is divided by 16 plus four. Divided by 16 is a fraction with a four in the denominator, the Y squared stays on top and 16 divided by 16 will get us that one that we need from my standard form. What we have is our 16. In the denominator of the X squared term would be my A squared. So a is plus or minus four and these are my ex intercepts of my lips cause they're under the X squared term B squared is four. So b equals plus or minus two. And these are my why intercepts cause they're under the Wye Square term. We last want to notice that this ellipse has a center at the origin 00 because of on Lee the excreting the wise great on the top of my standard form. I'm gonna go over to my grade here, and I'm going to sketch out my ex intercept points so I'll go to the right four and to the left, negative for and I'm going to go My y intercepts up to two and down to negative, too. I'll sketch the top half of my ellipse and the bottom half of my lips centered at the origin. This is the equation X squared plus four y squared equals 16 which remain into our standard form of our lips. And that's what this graph shows

So I'm gonna grab an equation for you. That is gonna be any lips. This particular problem has an X Plus three squared on top of 16. Plus why plus two squared over four equals one. We can see that it is an equation of an ellipse because it has that standard form with an X squared piece on top, a Y square piece on top, numbers on the bottom, plus on in the middle equals toe one. This particularly lips, um, is gonna be centered not at the origin 00 because of the plus three. And the plus two on the inside of the parentheses is gonna make this the lips centered at a negative three comma negative to as X and y values, we always take the opposite of the numbers that are in the parentheses to find our center. Now, with the standard form of our lips, the 16 in the four will give us a script and B squared values. A squared is 16 so a is plus or minus four, which makes an X intercept collection of points. If we were to be plotting on the access s I'm gonna quote ex intercepts were still gonna use the four and minus for. But Connor, as a distance away from the center value as we move out four units on those sides, the four under the Y squared piece is R B squared. So B squared equals four, which would make B plus or minus two. These would normally be what we call our Why intercepts if we would be centered on the, uh, access at the origin. But instead, we're going to use the plus or minus two just to get a movement away from the center. Two units in both directions, up and down. So what I'm gonna do is plot this. Ah, negative. Three negative. Two point 123 one two. So I'm down here and I'm gonna move four units to the right and to the left along the X values. So I'm gonna go to the right. 1234 into the left. 1234 at one. And negative. Seven, respectively. And then I'm gonna move up and down to 12 I'm on the access there. 12 down at negative four. So we have our lips graft. This would be the top half Come around for the bottom half and there is our equation. Graft X plus three squared over 16 plus why plus two squared over four equals one.


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