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Thread: [SOLVED] Analytic Geometry Q9

  1. #1
    Member looi76's Avatar
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    [SOLVED] Analytic Geometry Q9

    Question:
    A line passes through the points: $\displaystyle A(1,1)$ & $\displaystyle B(5,7)$. Find the coordinates of $\displaystyle P$ if $\displaystyle P$ is three times as far from $\displaystyle A$ as from $\displaystyle B$.

    Attempt:
    I know that I can find the points of $\displaystyle P$ using this formula: $\displaystyle X = x_1 + r(x_2 - x_1$) & $\displaystyle Y = y_1 + r(x_2 - x_1)$, but how can I find $\displaystyle r$?
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  2. #2
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    Quote Originally Posted by looi76 View Post
    Question:
    A line passes through the points: $\displaystyle A(1,1)$ & $\displaystyle B(5,7)$. Find the coordinates of $\displaystyle P$ if $\displaystyle P$ is three times as far from $\displaystyle A$ as from $\displaystyle B$.

    Attempt:
    I know that I can find the points of $\displaystyle P$ using this formula: $\displaystyle X = x_1 + r(x_2 - x_1$) & $\displaystyle Y = y_1 + r(x_2 - x_1)$, but how can I find $\displaystyle r$?
    There are two different points P. In my sketch one point is $\displaystyle P_{red}$ and the other is $\displaystyle P_{blue}$.

    1. $\displaystyle P_{red}$: Use proportions:

    $\displaystyle \dfrac{AP}{PB}=\dfrac31~\implies~\dfrac{AP}{AB-AP}=3$

    Solve for AP. I've got $\displaystyle AP=\underbrace{\frac34}_{\text{= r}} \cdot AB$

    2. If you use the equation of the line:

    $\displaystyle l:\left\{\begin{array}{l}x=1+r\cdot (5-1) \\ y=1+r \cdot (7-1)\end{array}\right.~\wedge r=\frac34~\implies~ \begin{array}{l}x=4 \\ y=\frac{11}2\end{array}$

    3. I leave the calculation of the coordinates of $\displaystyle P_{blue}$ to you. (For confirmation only: (7, 10))
    Attached Thumbnails Attached Thumbnails [SOLVED] Analytic Geometry Q9-zweipkte_auflinie.png  
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