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Zn744t"(4- ' Ueisncnyla'. !i ' 4' , ~JJJnial Mgorl mb} | i -Julg 4 Iv ~CTcni'cInterval:Center:Radius:...

Question

Zn744t"(4- ' Ueisncnyla'. !i ' 4' , ~JJJnial Mgorl mb} | i -Julg 4 Iv ~CTcni'cInterval:Center:Radius:

Zn 744t"(4 - ' Ueisncnyla'. !i ' 4' , ~JJJnial Mgorl mb} | i - Julg 4 Iv ~CTcni'c Interval: Center: Radius:



Answers

Find the center $C$ and the radius $a$ for the spheres. $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

In this equation. We in this problem, we're going to use the equation of the circle written in standard form where the center of our circle hk is shown here in the formula and a radius is in our formula square. So we can we can take care of that by taking the square root. So our equation is x squared plus y squared. Mhm minus X squared plus Y. Yeah, plus 1/4 equals zero. Mhm. Okay. Okay. So we're going to use an idea called completing the square to make our X terms. Who I have an error here. This is not an X squared is just an ex. Sorry. Okay. So we're going to make our X terms look like this format and we're going to make our Y terms. That's why square in this, Y look like this format. And we're going to get our constants are numbers to the other side so that we can use them to find our radius. So the first thing we need to do is rewrite our equations so that our X terms are together and we're going to leave some room to complete the square are why terms are together And we're going to leave some room to complete the square and we want our 1/4 to go to the other side. So we're going to subtract 1/4 from both sides. So our right side is now -14th. Okay, to complete the square on our X terms, we need to take half of the number in front of the X. The number in front of the X now is negative, one half of negative one is negative one half, I'm going to write that. There is a remainder. We want to square negative one half, that's negative one half times negative one half is going to be positive 1/4. If I add 1/4 to the left side of the equation, we know that we have to add 1/4 to the right side of the equation to keep it balanced. Okay now with our wise we one half of this positive one, which is one half, we're going to write it there as a reminder we're going to square it one half times one half is 1/4 and we're going to add that to both sides of the equation. Okay now we want to factor X squared minus x plus 1/4 and we just made it to where we know what the answer is. We need to numbers that multiply to be 1/4 and add to be negative one negative one half times negative one half is 1/4 and negative one half plus negative one half is negative one. So this factor is to be x minus one half times x minus one half. But we're going to write that is x minus one half squared because that's the format we're looking for. Okay. Using the same process for these wise this thing is going to be a factor to be why plus one half squared. We wrote that reminder there. We created it so that that would work. Our right side is negative. 1/4 plus 1/4 0 plus 1/4 is +14 So this is our equation in standard form. Now we can use that to find our center. Remember our center is hK we need these to be negative signs to be able to see that. So we're going to quickly rewrite ry term Instead of why? Plus 1/2 we're going to write in this y negative. Why minus negative one half square. Just pretend that that's up there. Okay now we can see that The X- is part of the formula sir, H is 1/2 and the y minus is part of the formula, So RK is negative one half. Yeah. Okay. And we know that our r squared is equal to 1/4 taking the square root of both sides. We have the square root of our squares are this would have been plus or minus, but we want the positive because we're talking about the difference between two points. So the positive square root of 14 is equal to the square to one over the square to four, which is equal to one half. So our radius is one half, our center is one half comma negative one half.

In this fusion the given data is that is the equation of the sphere is given, that is x squared plus y squared plus jerry square minus four. X plus four. White plus two, jail is equal to zero. No in the question it is saying that we have to find mhm. The ideas and uh center off his hair. So this is the occasion. No I am answering the execution. So in the first right given equation again that is X squared plus Y square plus jerry square -4 x plus four right plus two. Dad is equal to zero. No national step is rearrange it. So it will be a tennis. It is X squared minus four X plus why is square plus four? White plus jerry square plus two, dead Is equal to zero. No next is babies. We have to make a square in each variable. So it will be internet for making this square. It will be the niceties. X squared minus four. X plus four then plus Y squared plus four way. Yeah Plus four then plus Jerry Square plus to jail. Plus one is equal to zero plus four plus four plus one. Simplify it. Then it really written is that is this will become that is x minus two. Always square plus. This will become that is Y plus to all its square. And this will become that it scared plus one all the square and that is equal to now Ed right side that is Euro plus four plus four plus one. So four plus four will be 88 plus one. That is nine. So it means there is equal to nine. No next babies. Right? The extended equation of is where that is yes X minus X. Not only square plus y minus why not? Holy square plus administered not all square is equal to our square. So for the center of the standard equation of Israel will be tennis. That is X. Naught comma why not comma did not? And radius will be written as that is equal to are. So now I know my next step is compare the resulting equation with this standard equation. So after comparing I will get the center coordinator of center. That is that's not why not? You're not will return is that is two comma minus two, comma minus one. And the ideas will be done as that is The square root of nine. Or I can say that that is three. So this is as well given fusion. Thank you.

We need to write the equation of a circle given the center and the radius. So our equation of a circle is x minus h squared plus Y minus k squared is equal to r squared, H and K come from our center and our comes from our radius. So we have X zero squared plus why minus zero squared equals four times the square root of two squared. So this simplifies to be x squared plus Y squared Is equal to 32.

In this solution the given that it is that is the equation of despair. Given that is exit square plus Y squared plus jerry square mhm -4 X is equal to zero. No indication. It is saying that we have to find radius and uh center of is fair. So this is the occasion. Now I'm answering the execution so in the as well, first I will write the given equation again that is X squared plus Y squared plus jerry square -4. X is equal to zero. No message, tepees rearrange it So well written and said it's X squared plus four X plus Y squared plus jerry square. Sorry, X squared minus four X plus y squared plus jerry square is equal to mm deal No next day piece we have to make square in each variable completely square. So it will written as it is X squared minus four X plus four plus Why square Plus Jerry Square is equal to 0-plus four. So now again, since we've heard then it will be written as that is. This will become that is X managed to all the square plus then why square plus jerry square is equal to zero plus four will do that is four. So you can see that I had returned the each variable in square forum. So my next step is I am writing the standard equation of despair. So equation is that is x minus X. Not only square plus x minus X. Note it's not always square plus Y minus. Why not always square plus. They're managed not only square is equal to our square. There center will be written as that is X not form a Why not coma jade knob and radius is equal to our So use this extended equation of his fair and compared with the resulting equation, that is this is resulting equation. So after I will get center and areas. So I can say that after comparing the center will be the coordinator of center will be that is X not why not? They're not will beat. That is to than zero, then zero And release will be written as it is a square out of four. Or I can see that it will be you that is to. So this is a sort of given fusion that is centuries two comma zero, comma zero. And it is his studies to thank you.


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