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Plane $Z$ passes through the center of sphere $Q$.a. Explain why $Q R=Q S=Q T$b. Explain why the intersection of the plane and the sphere is a circle. (The intersec...

Question

Plane $Z$ passes through the center of sphere $Q$.a. Explain why $Q R=Q S=Q T$b. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any.plane passing through the center of the sphere is called a great circle of the sphere.)

Plane $Z$ passes through the center of sphere $Q$. a. Explain why $Q R=Q S=Q T$ b. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any.plane passing through the center of the sphere is called a great circle of the sphere.)



Answers

Find an equation of the sphere with center $ (-3, 2, 5) $ and radius 4. What is the intersection of this sphere with the $ yz $-plane?

So the way that we're going to do this is by using completing the square. So ahh. Completing the square here. Not too difficult to see. Hey, um, get something that looks like that. And then well, we just to write this stuff out and then we got nine is twenty one single toe five. Right. This stuff out right here. So the center that one too negative for in the radius of six.

So in these questions. So the question represent their spears basically this is x squared plus y squared plus b squared. But I guess man a six way six White and plus 2000 plus 17 ecological going to saw that this equals and represent spear and find their centers and radius. I. So we can clearly see that since the the coefficient of a square by square and square one. So we can see and uh there is no extra time, the only X, y and Z. To. So this the presenters pair Okay. And for the radio saying that for the regions and the centre we will find we will make a square. So this is it, that's a square plus old. So this will be like X plus pool is square, Okay, okay, so and there will be a 16 plus over here. So we will just subtract 16 for the balancing out over there plus why minus six y y minus of three square again, nine will be over here plus over nine minus nine. And yet. And plus to get that means jet plus one square one again will be some minus one over here plus 17 equal to the right, so it will be access exploits poor, it's quiet plus y minus of three Squire Plus jet plus of one square. And this is 16 and 9 25. And and it is 26. So this will be a minus of 26 and plus or 17. So it will be like Uh plus -9, that will be nine this side. So we can clearly see that this this will be equal to three is quite so we can clearly see that center have magnitude of the coordinator of this is x minus four minus of four and three and minus 100 G s r equals 23 squares of three. Okay, Thank you.

The problem is, if our tests go to X y Z A is equal to a one a two a three on being physical Toby one b two b three showed that the factory equation our minus a dot are minus B is equal to zero. Reference fans of the fear and find its center and writers first. Our mind say it's acquit you X minus the one uh, I minus two and C minus three. Our dynasty is for two X minus one. Why must into c minus B three is it? And since our minds a dot I'm speeds, you got zero. So we have X minus the one times X months be one us. Why months? A too Hamzawy minus p too. US C minus three. I have seen my species very. It's equal to zero. And then we have This is ex Claire minus. You want Pastilles? One Thanks us. The one you want us. Why square minus a two Class B too. Time's line us to be too us. The squire minus a three plus three. Come see plus three Very. It's equal to zero. Then we completed Squires. Have X minus a one plus B one must be one squire Us. Why minus et Tu asked me to over to squire plus C minus a three class B three over two square. It's me, conto. They want us to be one squire over for us to pass speech. You squatting over four US A three speed three square. We're four minus. They want you want one of A to be too. When I say three, this is he caught you by force times They want my spee one squire us A to minus B too. Squire us a three minus speed three square. So toe toe equation Armaments ate that are speaking. He's got zero represents A sphere on the center is they want class B one, four, two, two plus two over to a three. Plus the three over, too. Vitus is equal to one half routine stuff they want minus B one square us ain't You might speed too. Squire plus a three minus B three square

This time they say here, intersex the Why is he playing? So why Let me want to describe its trace. Okay, well, they mean the traces that intersect Now think about us here. It's nice. And round three dimensional. Actually, it's a nice hollow service. And no matter what, when you cut a plane through it, well, that ship look like Well, it's always gonna look a lot like a circle like this. Maybe it'll be quick. Are Maybe you're not cutting through it and half our imagine shifting this plane. Let's try cutting it a little higher in our circle appeared circle about a little small area. Well, now, what if we shift even higher? We're right here. Well, now we're only intersecting at a single point right here. Okay, It's ah, There are possibilities for the trace. It's going to be their circle. Our single point. Where do you get the biggest circle? Whoa! And you're cutting the spear and half because the maximum radius of any trace you get from a circle. It's just the radius of the spirit. Notice. We're sliding the plane up the radius of those circles for those traces are getting smaller and they collapse to a single point. Eventually


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