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Thread: Analytic last

  1. #1
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    Analytic last

    Coordinates of A is (1,1),Coordinates of B is (5,4).
    There is a point P moving over the line y=2x+4. When the sum of |AP|+|PB| is minimum with respect to P, find the ratio of PB to PA?

    My work:
    I found the symmetric of B. y=2x+4 so slope will be -1/2.
    y-4=-1/2(x-5)
    y=-x/2+13/2
    y=2x+4
    We find P as(1,6)

    I found PA and PB AS 5 and squareroot20.
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  2. #2
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    Re: Analytic last

    Quote Originally Posted by kastamonu View Post
    Coordinates of A is (1,1),Coordinates of B is (5,4).
    There is a point P moving over the line y=2x+4. When the sum of |AP|+|PB| is minimum with respect to P, find the ratio of PB to PA?
    Does the phrase "P moving over the line y=2x+4"mean that the point is on that line?
    If so then the line is $\{(t,2t+4) : t\in\mathbb{R}\}$
    So $|AP|+|PB|=\sqrt {{{\left( {t - 1} \right)}^2} + {{\left( {2t + 3} \right)}^2}} + \sqrt {{{\left( {t - 5} \right)}^2} + {{\left( {2t} \right)}^2}}$

    SEE HERE.
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    Re: Analytic last

    Yes it does.
    OK.B' is (-3,8)
    By using A, slope is -7/4
    Equation of the line is y=-7/4x+11/4
    I solved this with y=2x+4
    P(-1/3,10/3)
    I found PB/PA = 2.
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    Re: Analytic last

    Quote Originally Posted by kastamonu View Post
    Yes it does.
    OK.B' is (-3,8)
    By using A, slope is -7/4.
    Slope has nothing whatsoever to do with this question.
    We are asked to minimize the distance $A\text{ to }P$ plus the distance from $B\text{ to }P$: $|AP|+|PB|$
    The point $A=(1,1)$, the point $P=(t,2t+4)$ so the distance $|AP|=\sqrt{(t-1)^2+(2t+4-1)^2}$.
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    Re: Analytic last

    I also found the minimum distance. I also used distance formula.Book's answer is 1/2,my answer is 2, your answer is 1/3.
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    Re: Analytic last

    Quote Originally Posted by kastamonu View Post
    I also found the minimum distance. I also used distance formula.Book's answer is 1/2,my answer is 2, your answer is 1/3.
    My answer agrees with your book: $\dfrac{|AP|}{|PB|}=\dfrac{1}{2}$. SEE HERE
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    Re: Analytic last

    Did you find x=-1/3 as I did?
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    Re: Analytic last

    I apologise. I will check my solution again.
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    Re: Analytic last

    You posted another problem yesterday which had to do with finding distances between points and there, as here, you talked about the slope of a line and finding a "symmetric" point. Neither of these problems has anything to do with either of those, just the formula for the distance between two points.
    Thanks from Plato
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    Re: Analytic last

    Quote Originally Posted by kastamonu View Post
    Yes it does.
    OK.B' is (-3,8)
    By using A, slope is -7/4
    Equation of the line is y=-7/4x+11/4
    I solved this with y=2x+4
    P(-1/3,10/3)
    I found PB/PA = 2.
    Correct

    B'(-3,8) is the symmetric of B with respect to the line L: y = 2x + 4

    Equation of line AB': y = -7/4 x + 11/4

    Point P is at the intersection of (L) and (AB'): P(-1/3,10/3)

    so PB/PA = 2 or PA/PB = 1/2
    Last edited by Idea; Jun 25th 2017 at 12:52 AM.
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    Re: Analytic last

    Many Thanks.
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