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Math Help - coordinate geomatry

  1. #1
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    coordinate geomatry

    Determine the ratio in which the line 3X+Y-9=0 divides the line segment joining the points(1,3) and (2,7)
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  2. #2
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    Hello, nitinnitin2!

    Determine the ratio in which the line 3x+y-9\:=\:0 divides
    the line segment joining the points: A(1,3)\text{ and }B(2,7).

    The line through A and B has slope: . m \:=\:\frac{7-3}{2-1} \:=\:4

    . . and has the equation: . y - 3 \:=\:4(x-1) \quad\Rightarrow\quad y \:=\:4x-1


    The other line is: . y \:=\:-3x + 9

    They intersect when: . 4x - 1 \:=\:-3x+9 \quad\Rightarrow\quad x \:=\:\frac{10}{7} \quad\Rightarrow\quad y \:=\:\frac{33}{7}


    We have this diagram: . . \begin{array}{ccccc}A & & P & & B \\<br />
* & --- & * & --- & * \\<br />
(1,3) & & \left(\tfrac{10}{7},\tfrac{33}{7}\right) & & (2,7) \end{array}



    . . \overline{AP} \;=\;\sqrt{\left(\tfrac{10}{7}-1\right)^2 + \left(\tfrac{33}{7}-3\right)^2} \;=\;\sqrt{\left(\tfrac{3}{7}\right)^2 + \left(\tfrac{12}{7}\right)^2} \;=\; \sqrt{\tfrac{153}{49}} \:=\:\tfrac{3}{7}\sqrt{17}

    . . \overline{PB} \;=\;\sqrt{\left(2-\tfrac{10}{7}\right)^2 + \left(7-\tfrac{33}{7}\right)^2} \;=\;\sqrt{\left(\tfrac{4}{7}\right)^2 + \left(\tfrac{16}{7}\right)^2} \;=\; \sqrt{\tfrac{272}{49}} \;=\;\tfrac{4}{7}\sqrt{17}


    Therefore: . \overline{AP} : \overline{PB} \;\;=\;\;\tfrac{3}{7}\sqrt{17} : \tfrac{4}{7}\sqrt{17} \;\;=\;\;\boxed{ 3 : 4}

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