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Math Help - Locus Help

  1. #1
    Bar0n janvdl's Avatar
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    Locus Help

    We recently started with locus in Maths. Analytical Geometry to be precise, and I while I don't find it too hard, I do have a few questions about it.

    1. How do you know when exactly the locus is a circle, or a parabola, or a straight line, etc? Can someone give me a few tips on this?

    2. Are there any interesting things we can do with a locus? Or are they just used to find a point that is just as far from point A as point B...

    3. What is the plural of locus?
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    Quote Originally Posted by janvdl View Post
    We recently started with locus in Maths. Analytical Geometry to be precise, and I while I don't find it too hard, I do have a few questions about it.
    You have school in July?

    1. How do you know when exactly the locus is a circle, or a parabola, or a straight line, etc? Can someone give me a few tips on this?
    With analytic geometry.
    Let (0,0) be an origin. Let (x,y) be any point. And we want to find the locus of points equidistant from (0,0) 1 unit. So by the distance formula, \sqrt{x^2+y^2}=1 thus, x^2+y^2=1. Which is a circle.

    3. What is the plural of locus?
    Locii
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    Bar0n janvdl's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You have school in July?
    Yes we have a very short June holiday. Same with April and September. December holiday is about 2 months.



    [tex] With analytic geometry.
    Let (0,0) be an origin. Let (x,y) be any point. And we want to find the locus of points equidistant from (0,0) 1 unit. So by the distance formula, \sqrt{x^2+y^2}=1 thus, x^2+y^2=1. Which is a circle.
    How would i know when to use a hyperbole? Or a parabole?
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    Quote Originally Posted by janvdl View Post
    How would i know when to use a hyperbole? Or a parabole?
    Do you know the definition of what a parabola is in terms of locus?
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    Bar0n janvdl's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Do you know the definition of what a parabola is in terms of locus?
    No, they never teach us anything, we have to figure it out ourselves.
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    A parabola is the locus of points in the plane equally distant from a fixed line D called the directrix and a fixed point F, not on D, called the focus.
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    Bar0n janvdl's Avatar
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    OK, we were given a tutorial about locii and we were taught NOTHING about how to apply it like this... It also involves series and stuff.

    Please give explanations as well guys, thanks.

    Write 2,45 as a normal fraction, using  S_{\infty} .
    The ,45 repeats itself... Like 2,45454545...
    Last edited by janvdl; July 27th 2007 at 10:49 AM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by janvdl View Post
    OK, we were given a tutorial about locusts and we were taught NOTHING about how to apply it like this...
    I should hope not. Locusts are bugs, by the way. I think you mean either loci or locii, depending on who is teaching you.


    Quote Originally Posted by janvdl View Post
    Write 2,45 as a normal fraction, using  S_{\infty} .
    The ,45 repeats itself... Like 2,45454545...
    2. \bar{45} = 2 + \sum_{n = 1}^{\infty}45 \cdot \left ( \frac{1}{100} \right ) ^n

    The last term is a geometric series. And we know that
    S_{\infty} = \sum_{n = 0}^{\infty} ar^n = \frac{a}{1 - r}

    So
    \sum_{n = 1}^{\infty}45 \cdot \left ( \frac{1}{100} \right ) ^n = \sum_{n = 0}^{\infty}45 \cdot \left ( \frac{1}{100} \right ) ^n - 45 = \frac{45}{1 - \frac{1}{100}} - 45

    = \frac{45}{\frac{99}{100}} - 45 = 45 \left ( \frac{100}{99} - 1 \right )

    = 45 \cdot \frac{1}{99} = \frac{45}{99}

    Thus

    2. \bar{45} = 2 + \sum_{n = 1}^{\infty}45 \cdot \left ( \frac{1}{100} \right ) ^n = 2 + \frac{45}{99}

    = \frac{243}{99}

    -Dan
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    Forum Admin topsquark's Avatar
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    Personally I think the following way is both easier and faster than the series method.

    Let S = 2. \bar{45}

    Or
    S = 2.45454545454545 ...

    Then 100S = 245.4545454545 ...

    When we subtract the two:
    100S - S = 245.4545454545 ... - 2.4545454545 ...

    99S = 243

    S = \frac{243}{99}
    without having to memorize formulas and stuff.

    -Dan
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  10. #10
    Bar0n janvdl's Avatar
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    Sorry, i meant locii. What was i thinking?

    Thanks Topsquark.
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    Here is your homework.

    Let the focus be at (0,0) and the the directrix be y=-1.
    Find the equation of the parabola.
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    Bar0n janvdl's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Here is your homework.

    Let the focus be at (0,0) and the the directrix be y=-1.
    Find the equation of the parabola.
     4p(y - k) = (x - h)^2

    Set  4p = -1 then  p = - \frac{1}{4}

     -1(y - 0) = (x - 0)^2

     y = -x^2
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  13. #13
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Here is your homework.

    Let the focus be at (0,0) and the the directrix be y=-1.
    Find the equation of the parabola.
    Quote Originally Posted by janvdl View Post
     4p(y - k) = (x - h)^2

    Set  4p = -1 then  p = - \frac{1}{4}

     -1(y - 0) = (x - 0)^2

     y = -x^2
    The problem with your method is that (h, k) is the coordinates of the vertex point, not the focus.

    Here's the derivation. Perhaps this will be of value to you.
    F(0, 0) and directrix y = -1.

    We wish to find the locus of points such that they are equidistant from the focus and the line.

    Call the general point where this occurs (x, y).

    The distance from the focus (0, 0) and the point (x, y) is
    d_1 = \sqrt{x^2 + y^2}

    The (vertical) distance from (x, y) to the line y = -1 is
    d_2 = y - (-1)

    We set these two distances to be equal and this gives us a condition on x and y.

    \sqrt{x^2 + y^2} = y + 1

    Now solve for y:
    x^2 + y^2 = (y + 1)^2

    x^2 + y^2 = y^2 + 2y + 1

    x^2 = 2y + 1

    y = \frac{1}{2}x^2 - \frac{1}{2}

    This is your parabola.

    -Dan
    Last edited by topsquark; July 27th 2007 at 01:41 PM.
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  14. #14
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    .

    if point are all equidistant from x, it is a circle. if they are equidistand from x and y, it is a line.

    locii is plural of locus.


    thats about all i know
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  15. #15
    Forum Admin topsquark's Avatar
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    Another one: An ellipse is traced out as the loci of the sum of the distance between the point and two foci.

    To make sure my definition is clear (I didn't bother to look it up to find a good one) to find the equation for an ellipse with two given foci F1 and F2, propose a point (x, y) and set up the distance relations such that the sum of the distances between F1 and (x, y) and between F2 and (x, y) is some constant.

    (A circle is an ellipse such that the two foci are the same point.)

    -Dan
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