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Math Help - Equation of a locus

  1. #1
    Junior Member
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    Post Equation of a locus

    there,
    I was unable to solve the following questions, please help me.

    1) A is a point on the X-axis and B is a point on the Y-axis such that:

    4(OA) + 7(OB) = 20, where O is the origin. Find the equation of the locus of the midpoint P of Segment AB.

    2) Find the equation of the locus of the midpoint P of Segment AQ, if:

    A= (-1,2) and Q is a point on the locus y^2 = 12x
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) A(a,0), \ B(0,b), \ P\left(\frac{a}{2},\frac{b}{2}\right)

    4OA+7OB=20\Rightarrow 4|a|+7|b|=20.

    Divide by 2:

    4\left|\frac{a}{2}\right|+7\left|\frac{b}{2}\right  |=10

    Then the coordinates of P verify the equation 4|x|+7|y|=10

    The locus is a parallelogram formed by the lines:

    4x+7y=10

    -4x-7y=10

    4x-7y=10

    -4x+7y=10

    2) A(-1,2), Q\left(\frac{a^2}{12},a\right), \ P\left(\frac{a^2-12}{24},\frac{a+2}{2}\right)

    Then x=\frac{a^2-12}{24}, \ y=\frac{a+2}{2}

    a=2y-2\Rightarrow x=\frac{y^2-2y+2}{6}.

    The locus is a parabola.
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