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DiscoveR: Circle, Point, or Empty Set? Complete thesquares in the general equation $x^{2}+a x+y^{2}+b y+c=0$,and simplify the result as much as possible. Under what...

Question

DiscoveR: Circle, Point, or Empty Set? Complete thesquares in the general equation $x^{2}+a x+y^{2}+b y+c=0$,and simplify the result as much as possible. Under whatconditions on the coefficients $a, b,$ and $c$ does this equationrepresent a circle? A single point? The empty set? In thecase in which the equation does represent a circle, find itscenter and radius.

DiscoveR: Circle, Point, or Empty Set? Complete the squares in the general equation $x^{2}+a x+y^{2}+b y+c=0$, and simplify the result as much as possible. Under what conditions on the coefficients $a, b,$ and $c$ does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.



Answers

Find the center and the radius of each circle.
$$(x+a)^{2}+(y-b)^{2}=c^{2}$$

Okay, so we went to showed up. This equation represents a circle. So we want to write this in the general form of our circle to here. Let's move or ex terms. I know what to complete the square for X terms as well as for Or why terms. Let's move our constant to our exercise. So here, where X we have negative to over to get a far, too, that you put the one we're gonna add one here on and, um, for why that's for over two to the power of to that's equal to four. That's odd for here. So this is going to give us a X minus one. Depart two plus. Why plus to depart two is equal to four. Here we see that our center value is one. Commented to Andi, or radius is equal to what's the square root of four, which is equal to two

Okay, so we want to find our equation of our circle. So it's that removing our extend together I mean, what to complete the square here. Since we have two x turns, I don't have one. White ERM this needs to go. So here we have 1/2 to the power to, which is one of the four. So we have 1/4 and plus 1/4 here. Okay, So now we can practice into X plus 1/2 to depart to plus y squared is equal to 1/4. So based on this, our center is negative. One over to combat sterile on our radius. What X equal to the square root of 1/4. What you did with two one half.

Okay, so we want to find our equation of our circle. So here we want to rearrange or expellees and completely square on the same thing for our y value. Okay. And let's move or constant to our right hand side. So for X, we have to over two to the power to what you did with one who were adding one and for why you have 1/2 to the power of to which is equal to 1/4. So here we're adding. What over for? Okay, so now we concocted us into an X plus one to depart to plus why plus one over to the power of two is equal to 1/4. Looking at this, you see that our center value is h comma K. So that negative one common negative one half on our radius is equal to the square root of our right hand side, which is equal to one hat

When the bush sure that this equation is an equation of a circle and then fine the center R and the radius. So to begin the group, if there are more than one variables, we group them together. So there is only one X squared. It will remain on its own. But I have one squared plus six way. We're gonna be completing that square. So rooming here and bring the two over to the right side. That becomes negative too. So what we have we're going to add half the middle term squared. So six over to cleared give three squared, which is nine. Because we added nine to the left side of the equation. We must also add nine to the right site. So what we have, we can no factor. I'll perfect square here. It's a perfect square train, no meal. Which factors are to be why plus three cleared, which is equal to nine. Negative two, which is positives. Seven camp. So from here we can tell what the center of our circle is. The X coordinate is Cyril. The wire quarter negative negative three on a radio would be the square root of seven


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