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QUESTiONThe symmetric equation of normal line at the point (-7, 7, 7) to the surface x?+y2+22 = 4 is 0444-27-2+ 0842-55-4 oc '#8 - Y187-48 oo'17-947-4} oe...

Question

QUESTiONThe symmetric equation of normal line at the point (-7, 7, 7) to the surface x?+y2+22 = 4 is 0444-27-2+ 0842-55-4 oc '#8 - Y187-48 oo'17-947-4} oe957-4-97

QUESTiON The symmetric equation of normal line at the point (-7, 7, 7) to the surface x?+y2+22 = 4 is 0444-27-2+ 0842-55-4 oc '#8 - Y187-48 oo'17-947-4} oe957-4-97



Answers

Find all points on the surface $x^{2}+y^{2}-z^{2}=1$ at which the normal line is parallel to the line through $P(1,-2,1)$ and $Q(4,0,-1)$

In this problem will be finding the equation for the tangent plane on the part of metric equations for the normal line, given the given function at a given point. Now, in order to do that, we will be following this formula here for the tangent plane. So no, we will need the partial derivatives with respect to x, y and Z first and then evaluate under given point. So first, to find a partial derivative with respect to X, we will be treating y in C as constants. So here you have to it times why and we evaluate were excess negative three from the wise one that gives us negative six. Next, we will find a partial with respect to why X and Z are considered constants in this case. So here we have X squared in the second term, there's only see does that would be zero. And since X's negative three when we square that that gives positive nine. Finally we find a partial with respect to see and for the first part of the equation, that is zero. So departure would be negative. XY and tear. Izzy is negative two. So that evaluates to positive 16. So now we can follow our work formula for the tangent line. So if some X that would be negative six times the quantity x minus negative three said that becomes plus three this if supply just nine times Why minus one This except C which is 16 time see minus This absurdity is negative two. So that becomes close to equals. Zero that we can simplify here. Nearly six x minus 18 This nine wine minus nine a 16 c I was 32 equals zero Combating leg germs. We have negative six x last night. Why? For fixing Z is equal to negative but that this is the equation for detention Plea? No, but a parametric equations Aaron follow They said some formula. So we already have the information we need. So here we can see that X equal to X subzero just negative three plus our f sub x, which was negative six. That would be minus 60. Similarly, why Weiss observers one f sub y times t that will give us 90 thank Z would be equal to be zero, which is negative. Two. Let's accepts. See 16 t This would be Parametric equation

We're giving the equation of a surface and the point on the surface when we were asked to find Parametric equations for the normal line to the surface. At this point, the surface is described by the equation X squared minus eight X y Z plus y squared plus six z squared equals zero, and the point on the surface is 111 to find the parametric equations for the normal line to the surface. At this point, we need to identify first a point on the line. Call it P. Not. We can take Peanut to be simply our 0.111 Amy want to find a vector parallel to the line. This vector just parable line is going to be perpendicular to the surface at this point. So to find this vector, let's take partial derivatives to find the tangent vectors. At this point, we can use this to find a normal vector so I could use completing the square to write this as a function of X and y. However, a way to might be easier to see is by using implicit differentiation. So all differentiate both sides with respect to X, so you get two x minus Eat. Why? Z And then we have minus eight x. Why? Partial derivative of Z with respect to X plus 12 z Partial of Z with respect to X equals zero. So we get that part of derivative of Z with respect to X is equal to a Y Z minus two x over 12 z minus eight x y. And so we had at the point part of every bit of busy with respect to X 111 is going to be eight minus two over 12 minus eight which is six over four or three halves. We have that the partial derivative of Z respect the why we confined similarly using implicit differentiation we have that this is gonna be the same. Except for now we're switching the rules of X and y So you have that partner derivative of Z with respect to why is equal to eight x Z minus two. Why over 12 z minus eight x y And so we have the partial of Z with respect toe Why evaluated at 111 is equal to a minus two over 12 minus eight, Just 6/4 or three halves. And so we have that. A normal vector to the surface at this point is three halves, three halves, negative one. And we could multiply guy through by two to obtain another member vector. So we get and is equal to 33 of negative too. So with the normal vector in the points weaken right Parametric equations for the normal line as X minus one equals three T Why minus one equals three T and Z minus one equals negative to t, and this is our answer.

Okay, So for this problem, we are again looking Thio Find, um, a vector normal to the surface at the indicated point on. Then convert that to a heretic. A series of parenthood which so we are given Z is some function of X. Why, Um, that is equal to the natural log of three X squared plus seven y squared plus one. And the point given +000 Um, so we are going to find the normal vector which is given by which is given by a sea of axe at X and not well, I said no dizzy of want Z with respect to why an excel not accept not wise I'm not and negative one. And then we will use that to convert to parametric equations and that will be given by X minus X. I'm not peoples z with respect to X at excitement wise, I'm not times t why minus watts or not, which is to see with respect to why an excellent wife not times t and seeing my cities do not which is equal to negative t. So the first thing we need to do is take the partial with respect to x of this monster. So the when we're working with the natural log, we're going to use the chain rule here. And the first thing we're gonna do is we're going to, um, take the derivative of the natural log, which is gonna be one over all the junk inside. So that's one over three x squared plus seven y squared plus one. And then we're gonna chain roll the derivative of the stuff that's inside. So three X squared with respect to X will become six x seven y squared with respect to X will be zero and one with respect to X will also be zero. So this is gonna give me six X over three X squared plus seven y squared, plus one. Okay, now we're gonna plug in 000 into this and we're going to get, um, Let's see, on the top, we're gonna have a zero, because it's gonna be six times zero. Um, so don't matter what the bottom is. Ze of X will just be equal to that. 000 will just be able to zero, and then we're gonna do the same thing in life is you fly again derivative of the natural log. So we'll have to chain rule this. So it'll be one over all the other junk, all the inside junk, rather, And then chain rule the inside. So three X squared. Well, with respect to why that will be zero seven last word with respect to why that will be 14. Ah, and then one 14. Why, Sorry, 14. Why? And then ah, one will just be zero. So when we simplify here, we will have 14. Why? Over three x squared plus seven y squared plus one. And when I evaluate Z with respect to why at the given 0.0 Um, hope so. I didn't need to make that line. It'll be, um I forgot that. Why? It'll be 14 times why? Why Zero. So it'll just be zero. So now I am ready to go to my pear metrics and I will have x minus x of not which was zero. Zero t Why? Minus Weiss I'm not. Which was also 00 t and ze mine. Aziz, I'm not which was zero his negative t um and we are allowed to say since t is arbitrary, um, we could say that Negative T and T are the same thing since appear, we don't have any comparisons, right? These are both gonna be equal to zero. So we can say X equals zero y equals zero and the Z equals t. Now that so this would suggest that our equation points along the Z axis.


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