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Thread: Analytic geometry

  1. #1
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    Analytic geometry

    Coordinates of point A is (1,3) and B is (5,-3).There is a point P
    moving over line y=x+4. Find minimum value of |AP|+|PB|.
    I used the distance of a point to a line formula but I found the
    answer featuring a square root.Answer must be 6.
    Attached Thumbnails Attached Thumbnails Analytic  geometry-sexus.png  
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  2. #2
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    Re: Analytic geometry

    $|AP| = \sqrt{(x-1)^2+(x+4-3)^2}$
    $|PB| = \sqrt{(5-x)^2+(-3-x-4)^2}$

    The sum is:
    $\sqrt{2x^2+2}+\sqrt{2x^2+4x+74}$

    This is minimized when $x=-\dfrac{1}{7}$ and the result is 10, not 6.
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  3. #3
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    Re: Analytic geometry

    Btw, suppose $\sqrt{2x^2+4x+74}+\sqrt{2x^2+2}=k$

    Multiply both sides by the conjugate:
    $\sqrt{2x^2+4x+74}-\sqrt{2x^2+2}=\dfrac{4x+72}{k}$

    Adding these together and squaring both sides gives:

    $4(2x^2+4x+74)=\left(k+\dfrac{4x+72}{k}\right)^2$
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    Re: Analytic geometry

    To minimize PA+PB, P must be at the intersection of the line y=x+4 with the line BC
    where C is the symmetric of A with respect to the line y=x+4

    C(-1,5)

    B(5,-3)

    minimum distance = \sqrt{6^2+8^2}=10
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  5. #5
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    Re: Analytic geometry

    Many Thanks.
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  6. #6
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    Re: Analytic geometry

    Then we could use the symmetry of a point to a line and find C. Multiplication of the slopes will be -1. Then find CB by using distance formula.
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  7. #7
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    Re: Analytic geometry

    P will be(0,4) from the midpoint formula we get C.
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