Results 1 to 3 of 3

Thread: Tangential surfaces

  1. #1
    Sep 2009

    Tangential surfaces

    Hey, guys. So I'm about to have a test in my calc class and I'm working through the review that my professor handed out. I really need to know whether I'm doing some of these problems right before test day arrives, so if you guys don't mind, would you check my work and answers for some questions?

    Here's one: "Two surfaces are said to be tangential at a point P if (1), the point P is on both surfaces and (2), the two surfaces have the same tangent plane at P. Show that the surfaces $\displaystyle (2x^2)+(2y^2)-(z^2)=100$ and $\displaystyle z=(1/10)(x^2+y^2)$ are tangential at the point (8,6,10)."

    Here's my work:

    a) For $\displaystyle (2x^2)+(2y^2)-(z^2)=100$, I found fx, fy, and fz, which are 4x, 4x, and -2z.

    Then I plugged the points (8,6,10) into that, so f(8,6,10) = (32,24,-20).

    So the equation of this plane becomes $\displaystyle 32(x-8)+24(y-6)-20(z-10)=0$, which condenses to $\displaystyle 8x+6y-5z=50$.

    b) For $\displaystyle z=(1/10)(x^2+y^2)$, I went through the same process.

    fx, fy, and fz = (1/5)x, (1/5)y, -1

    f(8,6,10)= (8/5, 6/5, -1)

    Equation of the plane: $\displaystyle (8/5)(x-8)+(6/5)(y-6)-(z-10)=0$,
    which also condenses to $\displaystyle 8x+6y-5z=50$.

    Therefore, these two surfaces are tangential.

    Is this right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Aug 2006
    That does work. But there is a much shorter way.
    Show that the point is no both surfaces.
    Show that the two gradients at the point are parallel.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Sep 2009
    Hm... Okay, I see. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tangential Vector
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Aug 8th 2010, 06:49 AM
  2. Tangential Arc Problem
    Posted in the Trigonometry Forum
    Replies: 10
    Last Post: Feb 25th 2010, 01:50 PM
  3. What is the tangential acceleration?
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: Nov 11th 2009, 03:34 AM
  4. Tangential component help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Nov 17th 2008, 11:46 AM
  5. Replies: 1
    Last Post: May 23rd 2008, 03:44 AM

/mathhelpforum @mathhelpforum