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| | I am Shantae from Bottrop Altstadt. I am learning to play the Pedal Steel Guitar. Other hobbies are Homebrewing.<br><br>Feel free to surf to my web page :: [http://tempstreets.com/wiki/index.php?title=User:BookerT92qxmixl Basement Windows] |
| In [[Geometry]], a '''locus''' (plural: ''loci'') is a set of points whose location satisfies or is determined by one or more specified conditions i.e., 1)every point satisfies a given condition and 2)every point satisfiying it is in that particular locus
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| [[File:Locus Curve.svg|thumb|right|400px|Each curve in this example is the ''locus'' of a set of points that lie on any line defined as the [[Conchoid (mathematics)|conchoid]] of a circle centered at point ''P'' and the line ''l''. In this example, ''P'' is 7cm from ''l''.]]
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| ==Commonly studied loci==
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| Examples from plane geometry:
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| * The set of points equidistant from two points is a [[perpendicular bisector]] to the [[line segment]] connecting the two points.
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| * The set of points equidistant from two lines which cross is the [[angle bisector]].
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| * All [[conic section]]s are loci:
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| ** [[Parabola]]: the set of points equidistant from a single point (the [[focus (geometry)|focus]]) and a line (the [[directrix (conic section)|directrix]]).
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| ** [[Circle]]: the set of points for which the distance from a single point is constant (the [[radius]]). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a [[Circles of Apollonius|Circle of Apollonius]].
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| **[[Hyperbola]]: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
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| **[[Ellipse]]: the set of points for each of which the sum of the distances to two given foci is a constant. In particular, the [[circle]] is a locus.
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| ==Proof of a Locus==
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| In order to prove the correctness of a locus, one generally divides the proof into two stages:
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| * Proof that all the points that satisfy the conditions are on the locus.
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| * Proof that all the points on the locus satisfy the conditions.
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| == Examples ==
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| [[File:Locus apollonius.svg|thumb|(distance PA) = 3.(distance PB)]]
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| The locus of the points P that have a given ratio of distances k = d1/d2 to two given points.<br>
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| In this example we choose k = 3 , A(-1,0) and B(0,2) as the fixed points.
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| :: P(x,y) is a point of the locus
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| : <math>\Leftrightarrow |PA| = 3 |PB| </math>
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| : <math> \Leftrightarrow |PA|^2 = 9 |PB|^2 </math> | |
| :<math>\Leftrightarrow (x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2 </math>
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| : <math> \Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0 </math> | |
| :<math>\Leftrightarrow \left(x-\frac18\right)^2+\left(y-\frac94\right)^2=\frac{45}{64}</math>
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| This equation represents a [[circle]] with center (1/8,9/4) and radius <math> \frac{3}{8}\sqrt{5}</math>.
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| [[File:Locus3a.svg|thumb| Locus of point C?]] | |
| A triangle ABC has a fixed side [AB] with length c.
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| We determine the locus of the third [[Vertex (geometry)|vertex]] C such that
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| the [[Median (geometry)|medians]] from A en C are [[orthogonal]].
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| We choose an [[orthonormal]] [[coordinate system]] such that A(-c/2,0), B(c/2,0).
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| C(x,y) is the variable third vertex. The center of [BC] is M( (2x+c)/4, y/2 ). The median from C has a slope y/x. The median AM has a [[slope]] 2y/(2x+3c).
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| [[File:Locus3.svg|thumb|The locus is a circle]]
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| ::C(x,y) is a point of the locus
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| :<math>\Leftrightarrow</math> The medians from A and C are orthogonal
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| :<math>\Leftrightarrow \frac{y}{x} \cdot \frac{2y}{2x+3c} = -1 </math>
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| :<math>\Leftrightarrow 2 y^2 + 2x^2 + 3c x = 0 </math>
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| :<math>\Leftrightarrow x^2 + y^2 + (3c/2) x = 0 </math>
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| :<math>\Leftrightarrow (x + 3c/4)^2 + y^2 = 9c^2/16 </math>
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| The locus of the vertex C is a circle with center (-3c/4,0) and radius 3c/4.
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| [[File:Geassocieerde rechten.svg|thumb|The intersection point of the associated lines k and l describes the circle]]
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| A locus can also be defined by two associated curves depending on one common [[parameter]].
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| If the parameter varies, the intersection points of the associated curves describe the locus.
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| On the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is [[perpendicular]] to k. The angle between k and m is the parameter.
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| k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
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| ==See also==
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| {{Empty section|date=January 2014}}
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| ==References==
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| {{Reflist}}
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| *Robert Clarke James, Glenn James: ''Mathematics Dictionary''. Springer 1992, ISBN 978-0-412-99041-0, p. 255 ({{Google books|UyIfgBIwLMQC|restricted online copy|page=255}})
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| *[[Alfred North Whitehead]]: ''An Introduction to Mathematics''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-19784-2, pp. 121 ({{Google books|UyIfgBIwLMQC|restricted online copy|page=121}})
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| *George Wentworth: ''Junior High School Mathematics: Book III''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-15236-0, pp. 265 ({{Google books|cPlTB4qe40MC|restricted online copy|page=265}})
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| [[Category:Elementary geometry]]
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I am Shantae from Bottrop Altstadt. I am learning to play the Pedal Steel Guitar. Other hobbies are Homebrewing.
Feel free to surf to my web page :: Basement Windows