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Thread: Find All Numbers x

  1. #1
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    Find All Numbers x

    If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.
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  2. #2
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    Quote Originally Posted by magentarita View Post
    If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.
    1. Calculate the vector $\displaystyle \overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = ((x-(-3)), 3)$

    2. Calculate the magnitude of the vector $\displaystyle \overrightarrow{PQ}$ which has to be 5:

    $\displaystyle \sqrt{(x+3)^2+3^2} = 5$ Square both sides:

    $\displaystyle (x^2+6x+9) +9 = 25~\implies~x^2+6x-7=0~\implies~x=-7~\vee~x=1$

    3. Therefore $\displaystyle Q_1 (-7,4)$ or $\displaystyle Q_2(1,4)$
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  3. #3
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    great work.......

    Quote Originally Posted by earboth View Post
    1. Calculate the vector $\displaystyle \overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = ((x-(-3)), 3)$

    2. Calculate the magnitude of the vector $\displaystyle \overrightarrow{PQ}$ which has to be 5:

    $\displaystyle \sqrt{(x+3)^2+3^2} = 5$ Square both sides:

    $\displaystyle (x^2+6x+9) +9 = 25~\implies~x^2+6x-7=0~\implies~x=-7~\vee~x=1$

    3. Therefore $\displaystyle Q_1 (-7,4)$ or $\displaystyle Q_2(1,4)$
    Another great job here.

    Thanks
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  4. #4
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by magentarita View Post
    If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.
    Note that $\displaystyle Q\in\mathcal{C}$, where $\displaystyle \mathcal{C}$ is the circle centered at $\displaystyle P$ with the radius $\displaystyle r=5$.
    Just write the equation of $\displaystyle \mathcal{C}$ and substitute $\displaystyle Q$ to this equation and solve $\displaystyle x$ as earboth done.
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