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Math Help - Locus

  1. #1
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    Locus

    P(x,y) is a variable point on the line 2x-3y+6 = o, and the point Q divides OP in the ratio 3:2. Show that Q has co ordinates ( \frac{3}{5}, \frac{2/5) (x+3)) hence find the locus of Q.

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  2. #2
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    Hello deltaxray
    Quote Originally Posted by deltaxray View Post
    P(x,y) is a variable point on the line 2x-3y+6 = o, and the point Q divides OP in the ratio 3:2. Show that Q has co ordinates ( \frac{3}{5}, \frac{2/5) (x+3)) hence find the locus of Q.

    Thanks
    I think you mean that Q has coordinates:
    \left( \tfrac{3}{5}x, \tfrac{2}{5} (x+3)\right)
    By similar triangles (or using the formula that you'll find in this post), the point Q that divides OP, where P is (x,y), in the ratio 3:2 is
    \left( \tfrac{3}{5}x, \tfrac{3}{5}y\right)
    But if (x,y) lies on the line 2x-3y+6=0, then
    y = \tfrac23(x+3)
    So Q is:
    \left( \tfrac{3}{5}x, \tfrac{3}{5}\cdot\tfrac23(x+3)\right)
    i.e.
    \left( \tfrac{3}{5}x, \tfrac{2}{5} (x+3)\right)
    So, if the coordinates of Q are (h,k), then:
    h = \tfrac{3}{5}x and k = \tfrac{2}{5} (x+3)

    \Rightarrow x = \tfrac53h


    \Rightarrow k = \tfrac25(\tfrac53h+3)
    =\tfrac23h+\tfrac65
    \Rightarrow 15k = 10h + 18
    So the equation of the locus of Q is:
    15y = 10x +18
    Grandad
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