Question

At what point on the paraboloid y = x^2+z^2 is the tangent plane parallel to the plane x+2y+3z=1?

Answer 1

find the gradient of the surface

Answer 2

<2,-1,2>

Answer 3

grad f=<2x, -1, 2z>

Answer 4

yes

Answer 5

ok, now since we can't plug in anything that will change y, let's multiply the gradient by a scalar to get the j-component equal to 2 like it is for the plane

Answer 6

multiply by -2

Answer 7

right, and so we get?

Answer 8

grad f=<-4x,2y, -4z>

Answer 9

grad f=<-4x,2,-4z>
that y can't just appear from nowhere

Answer 10

so what value of x makes the i-components equal for the plane and grad f ?
what value of z makes the k-components equal?

Answer 11

x=1/2and z= 3/2 ???

Answer 12

from the equation as we have it we need x=-1/4 and z=-3/4

Answer 13

ok.

Answer 14

that's all then... boring problem

Answer 15

why qns with sphere have 2 sets of points?

Answer 16

hm... seems strange that y does not matter. Maybe I messed this one up :P

Answer 17

sorry? what do you mean?

Answer 18

for qns with sphere has 2 sets of points as ans why?

Answer 19

what do you mean it has 2 sets of points?

Answer 20

the ans fot this qn is (-1/4, -2,-3/4)

Answer 21

the way we did it it looks like y does not matter, so (-1/4,y,-3/4)
seems strange that y does not matter though, maybe I made a mistake... not sure

Answer 22

so the point for y is 0 or -1?

Answer 23

neither, the result says y can be anything, so just write y=y

Answer 24

isnt y a function if x and z to begin with?

y = x^2 + z^2; therefore y=1/4