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Math Help - Normals with no point

  1. #1
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    Normals with no point

    hi all,

    Im not too sure how to answer this question:

    Find the equations of the normals to the curve xy=3 which are parallel to the line 3x - y - 2 = 0

    obviously the equations need to be

    y=3x \pm c

    but how do you solve for c?

    presumably you need to find the gradient function ie
    y'=-\frac{3}{x^2}

    do you set

    3x-2=y'

    thanks
    sammy
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  2. #2
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    Quote Originally Posted by sammy28 View Post
    hi all,

    Im not too sure how to answer this question:

    Find the equations of the normals to the curve xy=3 which are parallel to the line 3x - y - 2 = 0

    obviously the equations need to be

    y=3x \pm c

    but how do you solve for c?

    presumably you need to find the gradient function ie
    y'=-\frac{3}{x^2}

    do you set

    3x-2=y'

    thanks
    sammy
    1. The given line has the equation y = 3x-2

    2. If the normal to the curve has the slope m = 3 then the curve must have the gradient m = -\frac13

    3. That means y' = -\frac3{x^2} = -\frac13~\implies~|x|=3

    4. The normals with m = 3 have to pass through the points P(-3, -1) and Q(3, 1).

    Use the point-slope-formula of the straight line to get their equations.

    5. The given line: green. The 2 normals in red:
    Attached Thumbnails Attached Thumbnails Normals with no point-senkrtang_aufgerade.png  
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  3. #3
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    thanks earboth

    why is it so obvious when your shown the steps .

    Using y-y{_1}=m(x-x{_1}), i see that c=\pm 8

    cheers
    sammy
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