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Thread: Equation of line

  1. #1
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    Equation of line

    Find the equation of the line through M (-1,2) with gradient m if M is the midpoint of the intercepts of the x axis and y axis

    equation of line = y - y1 = m ( x - x1)

    what next I am confused with the midpoint of axis.

    Thanks
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  2. #2
    Super Member Bacterius's Avatar
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    Hello Joel,
    your line passes through the point $\displaystyle M (-1, 2)$, so you know that $\displaystyle 2 = -m + p$ (using the standard equation $\displaystyle y = mx + p$)
    Now, let $\displaystyle x_1$ be the intercept of the y-axis, and $\displaystyle y_1$ be the intercept of the x-axis. Then we have$\displaystyle M \left (\frac{x_1}{2}, \frac{y_1}{2} \right )$. Thus we have $\displaystyle x_1 = -2$ and $\displaystyle y_1 = 4$. Since these are intercepts, we can get some more information :

    $\displaystyle 0 = -2m + p$ (using $\displaystyle y = mx + p$)

    Now recover the equation we found first :

    $\displaystyle 2 = -m + p$

    Solving the two equations simultaneously :

    $\displaystyle \left \{
    \begin{array}{l}
    0 = -2m + p \\
    2 = -m + p
    \end{array}
    \right.$

    $\displaystyle \left \{
    \begin{array}{l}
    2m = p \\
    2 = -m + p
    \end{array}
    \right.$

    $\displaystyle \left \{
    \begin{array}{l}
    2m = p \\
    2 = -m + 2m
    \end{array}
    \right.$

    $\displaystyle \left \{
    \begin{array}{l}
    2m = p \\
    2 = m
    \end{array}
    \right.$

    $\displaystyle \left \{
    \begin{array}{l}
    4 = p \\
    2 = m
    \end{array}
    \right.$

    $\displaystyle \left \{
    \begin{array}{l}
    p = 4 \\
    m = 2
    \end{array}
    \right.$

    And the equation of the line follows : $\displaystyle \boxed{y = 2x + 4}$

    Plotting this line indeed shows that it passes through $\displaystyle M(-1, 2)$, and that this point is exactly midpoint between the x-intercept and y-intercept of the line

    Don't hesitate to reply if anything seems confusing.
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  3. #3
    Super Member
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    Hi bacterius,
    After reading your first paragraph I accepted this as your proof of the x and y intercepts of the required line ie that the x intercept is -2 and y intercept is 4. So immediately y=2x+4. Was it necessary to go further. I did it by taking a straight edge and rotating it about M so that M was the midpoint between the x and y axis


    bjh
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  4. #4
    Super Member Bacterius's Avatar
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    Indeed, this is a shortcut, since we know the y-intercept we immediately know $\displaystyle p$, then we can use the point M (not the x-intercept, sorry typo) to solve for $\displaystyle m$, the gradient. I just did it the long way, I'm glad you noticed it was possible to go faster
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