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Thread: 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 $\displaystyle y^2 = 12x$
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) $\displaystyle A(a,0), \ B(0,b), \ P\left(\frac{a}{2},\frac{b}{2}\right)$

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

    Divide by 2:

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

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

    The locus is a parallelogram formed by the lines:

    $\displaystyle 4x+7y=10$

    $\displaystyle -4x-7y=10$

    $\displaystyle 4x-7y=10$

    $\displaystyle -4x+7y=10$

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

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

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

    The locus is a parabola.
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