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Consider the paraboloid z = X? + 2y2 and the plane z=X+y+27, which intersects the paraboloid in curve C at (4,3,34) as shown in the figure to the right. Find the eq...

Question

Consider the paraboloid z = X? + 2y2 and the plane z=X+y+27, which intersects the paraboloid in curve C at (4,3,34) as shown in the figure to the right. Find the equation of the line tangent to C at the point (4,3,34). Proceed by completing parts (a) through (d) below:(4,3,34)a. Find vector normal to the plane at (4,3,34). Choose the correct answer below:OA (1,-1,-1)(-1,-1,1)(4,3,34)(1,1,27)b. Find vector normal to the plane tangent to the paraboloid at (4,3,34). Choose the correct answer below:

Consider the paraboloid z = X? + 2y2 and the plane z=X+y+27, which intersects the paraboloid in curve C at (4,3,34) as shown in the figure to the right. Find the equation of the line tangent to C at the point (4,3,34). Proceed by completing parts (a) through (d) below: (4,3,34) a. Find vector normal to the plane at (4,3,34). Choose the correct answer below: OA (1,-1,-1) (-1,-1,1) (4,3,34) (1,1,27) b. Find vector normal to the plane tangent to the paraboloid at (4,3,34). Choose the correct answer below: 0A: (8,12,27) (-8, 12,1) (-8, 12,27) CD; (8,12,1) The line tangent to C at (4,3,34) is orthogonal to both normal vectors found in parts (a) and (b): Use this fact to find direction vector for the tangent line. Choose the correct answer below: 0 A (0,-7,4) (-7,11,0) (11, - 7,4) 0 D. (11,0,4)



Answers

Line tangent to an intersection curve Consider the paraboloid $z=x^{2}+3 y^{2}$ and the plane $z=x+y+4,$ which intersects the paraboloid in a curve $C$ at (2,1,7) (see figure). Find the equation of the line tangent to $C$ at the point $(2,1,7) .$ Proceed as follows. a. Find a vector normal to the plane at (2,1,7) b. Find a vector normal to the plane tangent to the paraboloid at (2,1,7) c. Argue that the line tangent to $C$ at (2,1,7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line.

We are given a surface s in our three defined by the equation X y squared plus two y z equals 16 in part A We're asked to find the normal back here to the surface for normal vector n We know from this section the formula for a normal vector to a surface to find implicitly by an equation in variables X, y and Z. Well, this is going to be equal to if we call f of x y z x y squared minus sorry plus two y Z minus 16 so that our surfaces defined by f of x y Z equals zero Then our normal vector is given by partial of F and the X direction I plus the partial derivative of F in the Y Direction Joe plus the partial derivative of F in the Z direction. Okay. And we find that the partial derivative of F with respect to X from our equation, this is why squared partially as an equity respect to why is two x y plus two z and the partial derivative of F with respect to Z is to y yeah, so we have that are normal vector and is equal to why Squared I plus two XY plus two z jay plus two y Okay, Now, in part B, pressed to find a tangent Plane H to the surface s at the point equals 123 Now, in order to find the tangent plane, we already know the normal to the surface at the point p and of P, this is end of 123 which from part A. You know, this is two squared or four I plus two times one times two or four plus two times three or six or four plus six or 10 jay plus two times two or four. Okay, so we have our normal vector At T. We know that this normal vector is also normal to our surface s at P. So an equation of our tangent plane h must be of the form two X plus five y plus two z. So I simply divided each of these coefficients by two. So divided by the greatest common divisor equals some constant C find See will substitute key into this equation. So we have it. C is equal to two times one or two plus five times two or 10 plus two times three or six, which is 18. Therefore follows that the Tangent plane H two s F p is defined implicitly by the equation two X plus five y plus two z equals 18.

The question way are given day I look soy excess square by square plus four sex, but most on the back. You find the question off engine plant, the Alex Roy at this point. Look, you elected. Be back. Find depend American question off the line that is normal. Develop side at the point home a home one and find the acute angle. You see what we have to find the attitude but that the engine plane at the 0.1 mix with explain play. Yeah, and now moving towards this solution meet again. Mhm By taking the partial that it takes off ethics and, like, as exact such attacks did not know that and can be used to extract big question or complaint. So product FX for two weeks That has four a five p o by a It is that there s Hey, So now let and what we call a tow 448 by cooperating with points off. Come on, come on. One such bad and God fought for Come on it on and start a life. Thank you. So for X minus, plus four by minus toe. Last eight X minus one. You know that is expressed by six. Now what about e? Pushing? Can get refusing on and connect such a there set of equations are normal active mind that existed stupid. It's not less f X equals with less 15 white list. Why not less F white? Basically to bless thine and yeah, yeah, what's party? And on the unit? But that game is supposed to be a student. How about you can use digital definition? Might be like Paul ST Paul's toe and I thought I wanted by more often in tow. That is but one toe, uh, from a coma, but divided by one square square back case by thanks. So a off 256 days or approximately a 25.6 degrees things is a collection. Okay, mhm.

The question way are given day I look soy excess square by square plus four sex but most on the back You find the question off engine plant, the Alex Roy at this point. Look, you elected. Be back. Find depend American question off the line that is normal. Develop side at the point home a home one and find the acute angle. You see what we have to find the attitude but that the engine plane at the 0.1 mix with explain play. Yeah, and now moving towards this solution meet again Mhm By taking the partial that it takes off ethics and like, as exact such attacks did not know that and can be used to extract big question or complaint. So product FX for two weeks that has four a five p o by a it is that there s Hey, So now let and what we call a tow 448 by cooperating with points off to the 01 such at and God fought for. Come on it on and start a life. Thank you. So for X minus, plus four by minus toe Last eight X minus one. You know, that is expressed by six. Now what about e? Pushing? Can get refusing on and connect such a there set of equations are normal active mind that existed stupid. It's not less f X equals with less 15 white list. Why not less F white? Basically to bless thine and yeah, yeah, what's party? And on the unit? But that game is supposed to be a student. How about you can use digital definition? Might be like Paul ST Paul's toe and I thought I wanted by more often in tow. That is but one toe, uh, from a coma, but divided by one square square back case by thanks. So a off 256 days or approximately a 25.6 degrees things is a collection. Okay, mhm.

So then parametric ization our teachers say it's equal to this at the point P notes as this. So I teeter then It's by on three and said it's so. So then we take the pasha derivatives of teacher and said so for our teacher you have this. So our twitter at the point P notes will be equal to minus three. I plus three square root of three J. Then our outset will be equal to K. So we fight the costs products of our data because our Z. And it's we have three square root of three. I plus three D. So then this implies that the sergeant's line it's I jeans. Helene is going to be we have three schools of three. I last three G. So guys we have X minus the points. That is three square roots of three divided by two. I less Why my nights And I divided by two G. Glad is said when it's so key equals zero then so this implies we have Squares of three x why equal to nine. So then that's the sergeant's plane. So then the Parma authorization the parliamentary opposition In place. We have extra vehicle to three say to tita. And why? To be equal to six say six. You have six six say squared. See it's A. So then this implies that our X squared let's Y squared will be equal tonight. Sign squared to tita less. This will be six. sign square teeter squeeze. So we have nine. You have nine for sign squared pizza. Cause square teeter for this then plus 36 say to the power for seems to So this thing it's equal to six. You have six say squared sita Which is equal to six. Why? Because this is why. So you have. This should be Y Sudan. This implies that X squared plus Y squared minutes six way Plus night will be equal to zero. So this implies that our ex quake bless. Why minus three square will be equal tonight as the find. Uh And so.


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