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Math Help - Distance between lines

  1. #1
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    Distance between lines

    I 'm currently working on my exercise, the question is

    Two lines
    L1: 3x + y - 4 = 0 and
    L2: 3x - y + 3 = 0 are given.
    P (h,k) is a point on the line x - y + 1 = 0.
    Suppose the distances of P from L1 and L2 are d1 and d2 respectively.

    (a) Express k in terms of h.

    >> I 've got it: k = h + 1

    (b) Express d1 + d2 in terms of h.

    >> That's easy as well.
    ie. 1/(10)^(1/2) (|4h + 3| + |2h + 2|)

    (c) When d1 + d2 is at the minimum, what is the value of h?
    Find also the value of d1 + d2 and the coordinates of P at this instance.

    >> This 's my problem. I got stuck and have no idea at all.


    Please kindly help me.
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  2. #2
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    Quote Originally Posted by ling_c_0202 View Post
    I 'm currently working on my exercise, the question is
    ...
    >> That's easy as well.
    ie. 1/(10)^(1/2) (|4h + 3| + |2h + 2|)

    (c) When d1 + d2 is at the minimum, what is the value of h?
    Find also the value of d1 + d2 and the coordinates of P at this instance.

    ...

    Hello,

    I didn't check your final result, I can only show you how to answer the last question:

    You've got a function with absolute values. Change it into a piece-wise defined function:

    |4h+3|+|2h+2|=\left\{\begin{array}{lcc}4h+3+2h+2=6  h+5,& h\geq -\frac{3}{4}\\ -4h-3+2h+2=-2h-1,& -1 \leq h < -\frac{3}{4} \\ -4h-3-2h-2=-6h-5,&h<-1 \end{array} \right.

    As you can see you've got 3 pieces of straight lines which are connected.

    The change of the slope from negative to positive indicates a minimum. This happens at h = -0.75

    I've attached a drawing of the graph of your function.

    A Happy New Year!

    EB
    Attached Thumbnails Attached Thumbnails Distance between lines-betrag_minimum.gif  
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