Question

let H(x,y,z) = x^2+y^2+2z^2, and S be the level surface H(x,y,z)= 4. find the coordinates of a point P on the surface S where the tangent plane to S is parallel to the plane 2x+4z=0.

Answer 1

ok I know that if i find the gradH = normal vector for tangent plane

Answer 2

\[gradH=2 x i +2yj + 4zk\]

Answer 3

and I also know that the planes normal is 2i + 4k

Answer 4

but I don't know whats next can you help me out @eliassaab

Answer 5

@Jhannybean can you help me please :)

Answer 6

@Hero can you help me please

Answer 7

@SolomonZelman can you help me please

Answer 8

@ganeshie8 can you help me with this one

Answer 9

heyy

Answer 10

:)

Answer 11

this is the question :/

Answer 12

You're done !

Answer 13

what do you mean?

Answer 14

vectors are parallel if the components are proportional

Answer 15

two planes are parallel if their normal vectors are parallel, yes ?

Answer 16

so what are the coordinates

Answer 17

yeah

Answer 18

i just know that the y coordinate is 0 but i don't know about x and z

Answer 19

normal of tangent plane to S : \(2 x i +2yj + 4zk\)
you want this parallel to the vector : \(2i + 0j + 4k\)

Answer 20

yes

Answer 21

so take the cross product?

Answer 22

just set the components equal toeach other

Answer 23

2i +4k = 2xi +2yj + 4zk ?

Answer 24

or 2i +4k = 4

Answer 25

sorry was on phone

Answer 26

its ok dont worry about it

Answer 27

i mean set the ratio of components equal to each other :

\[\dfrac{2x}{2} = \dfrac{4z}{4}\]

Answer 28

that gives \(x = z\)

Answer 29

and you know \(y = 0\)

Answer 30

use the equation of surface to solve for the points

Answer 31

so x^2 + 2(x)^2 = 4

Answer 32

yes!

Answer 33

so x = z = squareroot(4/3)

Answer 34

Yep!

Answer 35

wait you should get two points

Answer 36

negative and positive

Answer 37

\[x = z = \pm \sqrt{4/3} = \pm 2/\sqrt{3}\]

right ?

Answer 38

yeah

Answer 39

how can i know what point to pick

Answer 40

so the points would be :
\[(2/\sqrt{3}, ~~0,~~ 2/\sqrt{3})\]
\[(-2/\sqrt{3}, ~~0,~~ -2/\sqrt{3})\]

Answer 41

so both

Answer 42

there can be more than one tangent plane parallel to the given plane

Answer 43

yes

Answer 44

yeah thats right

Answer 45

thanks I see it