5

$5 x+12 y-10=0$ and $5 x-12 y-40=0$ touch the circle $C_{1}$ of diameter 6 . Lines If the centre of $C_{1}$ lies in the first quadrant, find the equation of the cir...

Question

$5 x+12 y-10=0$ and $5 x-12 y-40=0$ touch the circle $C_{1}$ of diameter 6 . Lines If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.

$5 x+12 y-10=0$ and $5 x-12 y-40=0$ touch the circle $C_{1}$ of diameter 6 . Lines If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.



Answers

In Exercises $41-46,$ find an equation for the circle with the given center $C(h, k)$ and radius $a$ . Then sketch the circle in the $x y$ -plane. Include the circle's center in your sketch. Also, label the circle's $x$ - and $y$ -intercepts, if any, with their coordinate pairs.
$$
C(-1,5), \quad a=\sqrt{10}
$$

Okay. This is problem number 46. We are given some information, and we have to write the equation of a circle on Graph it. The information was given to us. The center is three one half and the radius is equal to five. Now, Hopefully, you remember that this is the equation that we a formula that we use when we're trying to find the equation of a circle. With this information center and radius, you can find it online. You can find it in the book. Some people even memorize it because it's pretty close toe one of the standard equations. Anyway, I'm gonna go ahead and use this formula, So I'm gonna go ahead and say that Oh, H and K, that's the center of coordinates. Right? So it's gonna look the same as this. Some parentheses, a nice big plus some parentheses. Nice big square. There equals the radius squared. The radius is five. So this is 25 over here. Um, so we're gonna have x minus. And why minus and with her agent K. That's her H. That's okay. Right now, this is technically the equation. So you're done unless your teacher says Put in the standard for him. And if you do, then you foil out the left, these two different parentheses, and then you just kind of simplified. Okay, You use foil and you have to subtract and add some stuff, but it won't be a very big deal. I'm gonna leave it like this because they don't specifically say what form they wanted in. So therefore, this is acceptable because it is, by definition, the equation of a circle, and I'm going to sketch it. Oh, boy. Um, let's go over to the playing here. I have three and one half. So 3/3. 123 and then up a half. Wait. Was it up a half? Yes, it was. It was positive. So I'm gonna have my center right there. Okay. What was the radius again? I believe it was five. Yep. So I can actually count five from here to help my drawing. I can go up and go like 12345 It'll pass through there. I'm not gonna label that point because I didn't arrive at it. Algebraic lee. But visually, you could tell, uh and then 12345 We actually do know what this one's gonna be? Because we went up one half. Right now. We got over 12345678 So that would be eight for the and then one half that we can determine. Algebraic Lee organ. Just reason through it, let me label my center. Okay, But we need to find the intercepts, and we're gonna have a ton of them. Look at that. It's gonna go all through all those. So let's go ahead and go back and do some algebra to figure those out so we'll have them. Exactly. Um, do you remember that the Y intercept happens when X equals zero? It is true you're looking for the Y intercept that's going to be when you're X is equal to zero, you're gonna come up with some answers for why, so we can go ahead and solve for that and figure it out? Um, so we're gonna put in zero over here for X, right? So we're gonna have zero minus three, and this is gonna be we're going to solve it for why? Um, negative three squared nine plus this happy fellow. Subtract nine from 25 I believe it will get 16. Okay, so, uh, to solve for a while, let's take the square root of both sides. Because this is a perfect square. That's kind of handy. Um, when we take this period of something simple, like a 16, we could just say four, right. But truthfully, the square root of 16 could also be negative for because negative four times negative for 16. Right? So we really have a plus or minus situation. And then this could be why minus one half appear and give. This helps, um, room. So we're gonna have to do basically two versions of this one where why minus one half equals four and one where? Why minus one half equals negative four. So in this version, we add one half to both sides. We have y equals 4.5. And over here we should be add one half to both sides and we end up with negative 3.5 for that other coordinate. Okay, so remember, X equals zero, and our why ended up being this and this. All right, let's go on to the graph and will do 4.5 So here's zero. And in 4.5 would be 123 4.5. All right, let's get the other one on there. That zero negative. 3.50 negative. 12 3.5. Okay, now we have They want all intercepts, right? If any. And their pairs. So we're not done yet. We've got to find this one and this one over here. So we have another pair to find That's going to be when we're looking for the X intercepts now, right? So we knew when the y intercept that was one x was here. But when they're looking for the X intercepts, that's when our why is equal to zero. Okay, so we're gonna go through this whole process again, But just we're going to do when Why is zero. So let me shrink this down. So it's kind of like, out of the way. Keep this, you know, just to look at, but teeny tiny Okay, s. So I'm gonna go ahead and set. Why? Equal to zero for this one. So I have X minus three squared. Plus why, which is zero. So this becomes negative on half squared which is 1/4 right. One negative, one half times negative. One half is positive. 1/4 equals 25. Good. Oh, this is going to be, Ah, little bit more messy, but we could make it work. Let's just foil this out. X squared minus six X plus nine plus on fourth equals 25. Let's go ahead and bring all our constants, all our all our interviews here over the other side. So nine and a quarter. So what is 25? Minus nine and a quarter? If you subtract nine, you get 16. But you got to go smaller than that. So it's actually 15 and three quarters. Let's say 15.75 shall we? But this is still hanging out over here. How do we solve for this? But we're going to have to factor out on X X minus six equals 15.75 Okay, actually, you know it'll be a lot easier. Let's let's go ahead and do this differently. Kind of walked into that. Let's go to the subtract 1/4 off of both sides here. So we have X minus three squared equals 24.75 Okay. And then we're gonna take the square root of both sides. Will have X minus three equals plus or minus the square root of this nasty all number. Excellent. Eso We're gonna go ahead and just add three to both sides, so X is obviously not equal. Three plus. Remind us. Now many of you probably want to have a decimal number for this. You could do that if you want to. I guess we have a calculator. Let's just go out and do it, um, three plus or minus the square root of So let's just do three plus the square root of is it 24.75 24.75 24.75? Obviously, it's 7.97 It's around eight, but it's just going to put 7.97 is one of the answers and the other ex eagles is going to be when that is, uh, instead of being added, is gonna be subtracted. So now let's figure out what three minus the square root of 24.75 iss negative. 1.97 etcetera is a approximation negative 1.97 Okay, so I'm gonna go ahead and graph using these. I've got Oh, I should tell you what the points are just to mention this. So that means that why is zero in both these cases? Right, So these are the two coordinates of the lying triceps. Well, almost eight and almost negative, too. Almost eight. And almost negative, too. So almost eight, 2345678 Almost eight. And almost negative, too. And I could go back and rewrite thes. You should probably label these ones as well. Okay. All right. At this point, I'm ready to draw the starting circle. It's gotta pass through all these points as we go around in a big loop de loop here. So let's start off you on that one that I mentioned. So we go up and then we're coming down my thoughts or what helped me get it. Works? Act. If I tried to draw it all the way through with this program, it would look like an egg. And that is the graph of our circle. Thank you for waiting through the whole thing. Thanks.

Given the equipment of a circle X squared plus y squared equals five. We must find the venture under idiot. So you what? When we have the equation of a circle, we can tell what a centuries just by looking at a recording its here each. In case if I write my equation to resemble this a little more closely would have gotten X minus zero squared. This wine minus zero square was equal to discredit A by quick. Because this is what we have to square another to get five. Now we can see that our center is equal. Stood origin 00 on a radius is equal to the square root of by. And that is approximately 2.2. Okay, but not what We want to sketch this grab. Okay, We know the center is 00 on the radius. Is two going to So every point in the circle will be 2.2 units away from the center. So if estimate 2.2 to be roughly king room, right, you're yeah, down here. So this would be rough estimate of the graph of the equation X squared plus y squared equals. I

In problem 20. We have the equation for a circle. We have X squared plus 10 x plus y squared is equal to zero. We need to go ahead and sold for the center, the radius and then finally graphic. So in this case, I think it's always good just to write down wth East and erred equation of a Circle and that just imprints it in her memory of what we're looking for. And from that equation, our center is gonna be at H K. And then the radius is gonna be our So we're looking to get our first equation in the standard form that X minus X squared plus y minus case great is equal to our square. In order to do that, it looks like we're gonna have to complete this square because we have ex term in there. We don't really want that. So we have in Prince sees. That's right. X squared plus 10 x leave little space plus y squared equal to zero. So inside the prince seizes wherever completing the square and what we're gonna do is open this in blue. So we're gonna take after that. I'm gonna take that quantity and square it, and that's gonna be our value over on the right. So my graph disappeared there, but I could get that back. But we'll have, um after 10 is five square 25. So and we added 25 to the left hand side. You have to add 25 to the right hand side, which means that we will have X plus five quantity squared plus y squared on the left, equal to 25. And from that, then it's easy to find that her center is at H K, which is gonna be negative 50 and then our radius is gonna be five. Let me go ahead and get this graph back here and it just doesn't want to stay with me here here. So now that we have it up, then it's fairly simple in order to graft this. So we just go to negative 50 That's the center of our circle as a radius of five. So it's gonna go all the way up to here. It's gonna get here, here and there. So let's just do a little circle in orderto connect those points. This isn't looking like much of a circle. But with that, that graph is more just for understanding. The hard part is completing the square over here to get in standard form. And most importantly, our math backs up our graph because we see that her center is that negative 50 and we have a radius of five.

In problem 24. We have the equation of a circle X squared plus Y squared minus two X minus six Y minus five is equal to his ear, and we need to go ahead and grab this and identify the center and the radius. In order to do that, I want to go ahead and put this in standard form. This is X minus, each squared plus y minus K squared. And that's gonna be equal to R squared. So it already begin that process. I'm just gonna kind of rearrange the left hand side because I'm gonna have to complete the square. So I want to get rid of that ex term. That's all by itself. So I'm gonna have X squared minus two X gonna leave a little space to complete the square in. And then I'm gonna do the same thing with the wise. I'm gonna have y squared minus six y little space, and then I'm gonna do equals and move that five to the right hand side. So I'm gonna make that a positive five. So to complete the square's first look at the X is here, and I have the coefficient of the exes. Negative too. So that will become half of that is negative. One squaring it is one. And then for the wise half of negative six is negative. Three squaring That is gonna give me nine. Now, I added a one the left hand side that used to go over here and I added nine to the left hand side. So just need to add that to the right hand side. So what, that's gonna end up giving me on I simplify it down is gonna be X minus one squared. Plus why minus three squared equal to 15. So we don't quite have Ah, perfect square on the right hand side. But we can go ahead and identify our center, which is at H K, which in this case is 13 And then that radius is just gonna have to be the square root of 15. So now that we have that information, go ahead and graft this. So let's go over to the scrap on the right hand side. We'll grab it in red. Their center is at 0.13 and then screw to 15. What's just a little bit less than four. So I like to graph out just four little points here to kind of use is a guide when I'm graphing and then I'll connect them with the best circle I can kind of do a quick sketch. Circles are pretty hard, but now that I have that, even if my graph is a little sketchy, um, that's really just to show that I understand what the math on the left hand side is trying to represent. The precision comes from our numbers and calculations on the left hand side or identified the center as 13 and the radius as Route 15.


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