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  1. #1
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    locus problem

    p311 q12
    question : a variable line passing through the point (5,0) intersects the lines 3x-4y=0 and 3x+4y=0 at H and K respectively. Find the equation of the locus of the mid-point of HK.
    thanks in advance
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  2. #2
    Moo
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    Hello,

    Quote Originally Posted by afeasfaerw23231233 View Post
    p311 q12
    question : a variable line passing through the point (5,0) intersects the lines 3x-4y=0 and 3x+4y=0 at H and K respectively. Find the equation of the locus of the mid-point of HK.
    thanks in advance
    Let H(x_H, y_H) and K(x_K, y_K)

    The equation of the variable line is : y=ax+b

    But we know that it's passing through (5,0).
    Thus 0=5a+b
    --> b=-5a

    Therefore, the equation of the variable line is : \boxed{y=ax-5a}, a is the variable

    So we know that :
    \begin{aligned} y_H & = & ax_H-5a & (1) \\ y_K & = & ax_K-5a & (2) \end{aligned}
    Because H and K are on this line.


    Plus, H is on the line of equation 3x-4y=0
    And K is on the line of equation 3x+4y=0
    ---> \begin{aligned} 3x_H-4y_H & = & 0 & (3) \\ 3x_K+4y_K & = & 0 & (4) \end{aligned}

    By substituting (1) into (3), you will have x_H with respect to a.
    Replace this expression of x_H in (1) to have y_H with respect to a.

    Do the same for x_K and y_k.



    Then, the midpoint of HK is M \left(\frac{x_H+x_K}{2}, \frac{y_K+y_K}{2} \right), which will be coordinates with respect to a

    Hope that helps... If you have questions about some steps, don't hesitate ^^
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  3. #3
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p311 q12
    question : a variable line passing through the point (5,0) intersects the lines 3x-4y=0 and 3x+4y=0 at H and K respectively. Find the equation of the locus of the mid-point of HK.
    thanks in advance
    In addition to Moo's excellent explanations I've attached the drawing of the locus and for your confirmation the equation of the locus:

    \frac{\left(x+\frac52\right)^2}{\left(\frac52\righ  t)^2}- \frac{y^2}{\left(\frac{15}{8}\right)^2}=1
    Attached Thumbnails Attached Thumbnails locus problem-locusproblem.gif  
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