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Find an equation for the sphere whose certer and radius are given Center RadiusChoose the correct equation of the sphere 0 A 65) " 6h 16 81 08 (2)+(4 (X+ 3)2 (...

Question

Find an equation for the sphere whose certer and radius are given Center RadiusChoose the correct equation of the sphere 0 A 65) " 6h 16 81 08 (2)+(4 (X+ 3)2 (2)7 (2+2)2 16 81 OD (x+ 312 6)6

Find an equation for the sphere whose certer and radius are given Center Radius Choose the correct equation of the sphere 0 A 65) " 6h 16 81 08 (2)+(4 (X+ 3)2 (2)7 (2+2)2 16 81 OD (x+ 312 6)6



Answers

Complete the squares to find the center and radius of the sphere whose equation is given. $$x^{2}+y^{2}+z^{2}+2 x-6 y-10 z+34=0$$

Were given the equation of a sphere, and we're asked to find the center and radius of this sphere. The equation is four x squared plus four y squared plus four z squared, minus four x minus 32 y plus eight C plus 33 equals zero. To find the center and radius, I need to write this equation in standard form. And to do that, I need to use completing the square. So first I'll group like terms. We have four X squared minus four X plus four y squared, minus 32 y plus four z squared plus eight z plus 33 equal zero Next al factor at the leading term from each polynomial and complete the square. So we have four times X squared minus X plus negative one half squared or 1/4 plus four times y squared minus eight y plus negative four squared or 16 plus four times Z squared plus two z plus one squared or one. This is equal to negative 33. For balance, I'll add four times 1/4 or one and four times 16 as well as four times one or four. So, actually, this right hand side. This is negative. 33 plus one is negative. 32 or negative four times eight plus four times 17. This is the same as four times nine. And so the standard form, if I divide both sides by four, is X minus one half squared plus y minus four squared plus Z plus one squared equals nine and so the center of the sphere is located at one half for negative one, and the radius of the sphere is the square root of nine or three.

To solve this problem, we're going to uh complete the square to get into this form. So we're starting off with X squared. And let's group all of our access together. So we're gonna have x squared minus two X plus y squared um plus six Y plus z squared plus eight D. Um equals negative one. And we want to complete the square. So we're going to um make this acts minus one squared. Kind of Plus one has to go here and this becomes X. Class three Squared, This becomes plus nine and this is um Z plus four squared And that becomes like 16. Then what we're left with is 25. So we see that the center is 1 -3 -4 with a radius of five.

We want to complete the squares here so we'll have X squared. That's eight X. That's why I squared. I asked for why plus Z squared -22 Z Equals and negative 77 completing the square. This will be express for Yes The U. s. 16. This is excellent for spread. That is This will be Y -2 spurred and that'll make this plus four and then this will be a Z minus 11 Squared which makes this plus 121. Right? Mhm. So um that's going to end up giving us 64. Based on that we have a radius of eight and the center of the sphere is For a negative 4- 11. That's our final answer.

So in this problem, were asked to find the center and radius of a sphere. Given that this fear satisfied by the following equation X squared plus y squared with Z squared this eight x minus six y plus twos, even 17 equals zero. So what we can do to show this is we can complete squares in the equation for X, y and Z because we don't have any cross product terms like X y or Y Z or X y Z, for instance. So when we complete the square in each of the variables, then we're going to get after simplification. We're going to get X Plus four squared plus y minus three squared Z plus one squared is equal to nine, which can be shown is equivalent to the problem statement. Once you expand all the terms, and this is the equation of a sphere with center given by the coordinates. Negative four three negative one and the radius is given by Square Root of nine, which is three units


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