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Learn about the general form of the equation of a circle (𝑥 − 𝑎)² + (𝑦 − 𝑏)² = 𝑟², where (𝑎, 𝑏) is the coordinates point of the centre of the circle and 𝑟 is its radius. Use this knowledge to help draw graphs of circles given their equation.
So the equation for a circle will always be in the form 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared equal to 𝑟 squared, where the center of the circle is 𝑎, 𝑏 and the radius of the circle is 𝑟. So to be able to graph any circle, we need to get it into this standard form for the equation of the circle. And let’s just take a second again to look at what the center is, what the radius is, and how it relates to the equation. So we can see that the equation has got 𝑥 minus 𝑎, but then the center is just 𝑎. So whatever the sign is inside the parentheses, we will swap it to give us the 𝑥-coordinate of the center. And the same with the 𝑦-coordinate. The 𝑥 is always next to the 𝑥, surprisingly. And the 𝑦-coordinate is always next to the 𝑦. So we’ve looked at the center there. Now taking a second to look at the radius, we are always given the radius in the equation as 𝑟 squared, but the radius is just 𝑟. So whatever we’re given in the equation, we must square root to give us what the radius is.
So looking at this graph, we must graph 𝑥 squared plus 𝑦 squared equals one. So you can see it’s not quite in a form that we want it in, to be able to draw it. But we can see that we are not adding or taking away anything to either 𝑥 or 𝑦. So actually, we could rewrite it as 𝑥 minus zero all squared plus 𝑦 minus zero all squared equals one. That means exactly the same as the equation we were given first of all. So that tells us that the center of this graph is going to be zero, zero. And the radius is always the square root of what we’re given in the question. Well the square root of one is just one, so the radius is one.
This is a very special circle; this is called the unit circle. And let’s start plotting it. So I’ll do a dot where the center is. And we know that the radius is one, so that means we can go one away from the center in every direction. First of all, up, to across there, down, left, and then right. So when we’re sketching it, we need to do each corner at a time. So start from wherever you want. I like to start from the top, and then go across to the next point, and then work away all the way around. Well see, this isn’t looking great so I’m gonna get one that I’ve done earlier, and it should look a bit like this. So obviously, we’re not going to be able to draw perfect circles. So for the answer to be correct, it just has to have the points correct for the center and the radius, and then also some kind of attempt at a circle.
So now let’s graph this circle 𝑥 plus two all squared plus 𝑦 minus two all squared equals sixteen. So we can quite clearly see that the center is going to be negative two in the 𝑥 as we must swap the sign. And two for the 𝑦-coordinate again because we must swap the sign. And the radius is going to be the square root of the number we’re given in the question. So the square root of sixteen, which we know is four. So first off, we will mark the center, which we know is negative two and two. And then we’re going to count four in each direction. So first of all, up, we will count four and mark. Then we’ll go down four, from two we’d expect that to be negative two. We’ll go left four, so that will be on negative six. And right four, so that’ll be on two. And then joining from point to point with a curve, going all the way around. You should get something like this. Now for our last question, we’re going to use completing the square with our knowledge of circles, to be able to plot one.
So we must graph the circle 𝑥 squared plus 𝑦 squared plus six 𝑥 minus twenty 𝑦 plus eighty-four. If you remember, we have to have it in the form 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared equals to 𝑟 squared. So we’ve got a little bit of work to do. The first thing we want to do, is to put the 𝑥s next to each other and put the 𝑦s next to each other. And now we’ve done that. We can look back at the general form for the equation of a circle, and we can see that that should be reminiscent of what something looks like when we completed a square. So we’re going to complete the square for the 𝑥-values and the 𝑦-values individually. First of all, 𝑥.
Remember, we need to half the coefficient of the 𝑥 by itself, which will be three. So that will give us 𝑥 plus three all squared and we’ll subtract from that three squared, which we know is nine. And we’re done with the 𝑥s. So moving on to the 𝑦s, we’re gonna do exactly the same thing. We’re gonna put 𝑦 minus half of twenty, so ten, all squared. And we’re going to subtract from that ten squared, which we know is a hundred. And then finally, from before, we’ve got plus eighty-four, equals zero. So collecting the numbers together, we’ve got negative a hundred and nine plus eighty-four, which gives us an answer of negative twenty-five. And then if we add twenty-five from both sides, we will get 𝑥 plus three all squared plus 𝑦 minus ten all squared equals twenty-five.
And now we’ve done the hard part. All we need to do is go back to what we know about graphing circles. So we need to find the center, which we can find by swapping the signs on each of the numbers in the parentheses. So the 𝑥-coordinate will be negative three and the 𝑦-coordinate will be ten. And then the radius is going to be the square root of twenty-five, which we know is equal to five. And then now we just need to graph it as we did before. So we’ll mark the center at negative three, ten. And we’ll count five up, down, left, and right, which on these axes will be two and a half squares. And then joining up as we did before, we should leave ourselves with a nice circle that looks a bit like this.
So, as I said, this is probably one of the hardest types of questions that you’ll get, when just asked to graph a circle. So in this case, all we needed to do is, although it look quite daunting, use knowledge we already had of completing the square. In the previous cases, we just needed to get it into its general form, so we could easily find the center and the radius of the circle, to be able to graph it.
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