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

1. TuringTest

find the gradient of the surface

2. mathstina

<2,-1,2>

3. TuringTest

4. mathstina

yes

5. TuringTest

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

6. mathstina

multiply by -2

7. TuringTest

right, and so we get?

8. mathstina

9. TuringTest

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

10. TuringTest

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?

11. mathstina

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

12. TuringTest

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

13. mathstina

ok.

14. TuringTest

that's all then... boring problem

15. mathstina

why qns with sphere have 2 sets of points?

16. TuringTest

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

17. TuringTest

sorry? what do you mean?

18. mathstina

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

19. TuringTest

what do you mean it has 2 sets of points?

20. mathstina

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

21. TuringTest

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

22. mathstina

so the point for y is 0 or -1?

23. TuringTest

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

24. amistre64

isnt y a function if x and z to begin with? y = x^2 + z^2; therefore y=1/4