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Math Help - Tangency Question

  1. #1
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    Tangency Question

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

    I just need some help getting started.
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  2. #2
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    Quote Originally Posted by lllll View Post
    At what point on the paraboloid  y=x^2+z^2 is the tangent plan parallel to the plane x+2y+3z = 1.

    I just need some help getting started.
    start it just like you would have in Calc one or two with a curve...set up your tangent plane equation
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  3. #3
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    Quote Originally Posted by lllll View Post
    At what point on the paraboloid  y=x^2+z^2 is the tangent plan parallel to the plane x+2y+3z = 1.

    I just need some help getting started.
    A normal to the given plane is i + 2j + 3k.

    A normal to the paraboloid is \frac{\partial y}{\partial x} i - j + \frac{\partial y}{\partial z} k = (2x) i - j + (2z) k.
    A normal of the tangent plane at the point (a, a^2 + b^2, b) is therefore (2a) i - j + (2b) k.

    You want to find the point at which the tangent plane has a normal parallel to i + 2j + 3k. You therefore want the values of a and b such that (2a) i - j + (2b) k = \mu (i + 2j + 3k) where \mu is a scalar. So solve the following equations simultaneously:

    2a = \mu ... (1)

    -1 = 2 \mu ... (2)

    2b = 3 \mu ... (3)
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