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4- Find equations of the tangent plane and normal line to the surface z2 4y2 22 = 9 at the point (5,-2,0)....

Question

4- Find equations of the tangent plane and normal line to the surface z2 4y2 22 = 9 at the point (5,-2,0).

4- Find equations of the tangent plane and normal line to the surface z2 4y2 22 = 9 at the point (5,-2,0).



Answers

Find equations of the tangent plane and normal line to the surface at the given point. $$z^{2}=x^{2}-y^{2} \text { at }(5,-3,-4)$$

It is prevalent Z is equal to the skirt, uh, excreted sliced bread. And, uh, if we're going to treat this as a level surface of a function later normal lines intendant planes, let's get the Z on the other side are functional baby expression on the weight, right? Always. I really waited at the point very negative for and find. So let's see the great in that case would be X over the square root of X squared plus y squared pressure is expected. Why it would be lying over this Gregory of X squared, plus my screen part of every respect. Dizzy, just negative one. And I want to evaluate at that 0.3 negative for well, on the bottom, we're gonna end up with that scene. So he squared plus four squared square ead just five So three ever find and they get it for over five and negative one. So that's the normal vector. Okay. And now we have been seeing the normal line will be a dramatized function. What's the initial points are three. Then watch the 3/5 of some parameter negative for minus 4/5 of some parameter C five minutes. That parameter okay? No, the tangent point. I don't see me. And the boys. Let's see. Partial of X 3/5 X minus three. Some try it forfeits. Why? Plus one. And then let's see, It's attract one. Uh, the minus wife. Okay.

And this problem. We're trying to find the equations. We're presenting attention plain and a normal line for their given equation at a given point. So first, what we will do is find the partial derivatives with respect to X online. So the partial derivative with respect to X would be equal to for a little why on a partial juror they respect why would be equal to negative for act over y squared. So at our first point, we're going to evaluate the partial derivatives. So here we have four over to which is equal to two. And and some worry at 2.12 would be equal to they could do or terms warn, go over to square. And that is equal to negative. Uh, there he refined that the equation representing the tension playing would be equal to two close to turned the quantity minus one minus one times a quantity. Why minus two the correspondent normal line would be given by X equals one us to tea y equals two minus warranty. Z is equal to tu minus. So here we have re equations representing the tangent clean under normal line for the second point going to evaluate no partial derivatives, so that gives more driven by four is one when the why we have negative or some negative born writing by four squared and then use poverty. Poor was 16 which in advice to 1/4. So here we have that the equation for the tangent clean would be equal to negative. Born this one times X minus negative one becomes a plus one plus 1/4 times Why minus four the corresponding normal line will be given by X equals negative one trustee one equals four plus 1/4. T and Z is equal to negative one minus. So here are tangent plane equation on the normal line equation.

In discussion, we record about a formula to compute attention plan. So it has a formula credit on the F dot with the X men X zero Y minus 50 Z manner disease zero must equal zero in this question. Were given the question Z echoed to the x square minus y square. When we had upon three 25 first I want to create a function F X. Y. And easy to decode. Do I bring the city to the other side? So we have the X squared minus Squire, Squire one Z. Therefore I can compute the caribbean on the F. Here we go to the two X -2 Y -1. Therefore we're blocking the point we have here and we go ju actually what you three? So we have the sixth one. Would you choose? So I had a minus far and secret u minus one. Then far we can write down the equation of the pension plan. It will be 6 -4 -1, dealt with the x minus three Y minus to the minus five. Coaches are all if we taught them we should get six eggs minus 18 then minus four. Y plus eight minus c plus five equals zero. If we simplify this one, we have the six x minus four y minus C. We have here will be minus 10 minus five echo +20. This will be the pension plan we're looking for now. We also need to fight the symmetric the question. So it has a farm xmas extra of F X. Equal to the y minus Y zero of F Y equal to z minus z zero of f set. They found a question we have here will be x minus three over six, ico dy minus two of minus form ego to the minus five of minus one. So this will be the symmetry question we're looking for.

In discussion, we record about a farmer that you can be a redundant plan. So it has a formula creating on the F. Dealt with the vector x minus x zero, y minus y zero, Z minus zero. And the point. So here we are given the question, Z is equal to the x squared minus square square and a 0.3 25 here the first time I want to create the function F, x, y and Z. And we bring the Z on the other side. So I have the x squared minus y squared minus Z. From here. And we'll be able to find a credit under F. It will echo at you That you X -2 Y -1. And from here I can compute a credit on the F evaluate and the point here. So we will have echo 26 -4 -1. And from here we can apply the formula to compute attention plan. So we have the six minus four minus one. We thought with x minus three by minus to the minus five equal to zero. Sorry, this one we have to send equal to zero and then it was 75. We have the six x minus 18 minus four. Y plus eight minus z plus five echo +20. It was 75. We have the +56 x minus +54 Y minus z minus five echo +20. This will be the attention plan we're looking for also we need to find a symmetric question. So it has a farm x minus x zero of uh f x echo two y minus y zero of f Y. You go to z minus z zero of the fz. Therefore it will blind the value. We have the x minus three of uh six echo, G y minus two of minus four equals z minus five minus one. So this will be the symmetric key question we're looking for.


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