First Definition and Locus Construction of a Hyperbola
2020-03-10 00:42
This interactive example dynamically demonstrates the first definition of a hyperbola, which is the locus of points in a plane where the absolute difference of the distances to two fixed foci is a constant. Using GeoGebra's geometric construction tools, it builds the foci, a moving point, and corresponding segments, utilizing the Locus command to generate both branches. This model intuitively illu
In collections
Secondary Vocational Mathematics
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="d131212c-8b1f-4fcf-9dad-33aaf22a027c" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1350" height="561" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="286" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1342" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="0" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1341" height="560"/>
<coordSystem xZero="680.8664709196419" yZero="307.666216522974" scale="19.269640116935108" yscale="19.26964011693516"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="c">
<value val="5"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<slider min="0.1" max="10" absoluteScreenLocation="true" width="200" x="74" y="89" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<caption val="$\large %n=%v$"/>
</element>
<element type="numeric" label="a">
<value val="4"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<slider min="0" max="c" absoluteScreenLocation="true" width="200" x="73" y="42" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<caption val="$\large %n=%v$"/>
</element>
<expression label="F_1" exp="((-c), 0)" type="point" />
<element type="point" label="F_1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-5" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large %n%v$"/>
</element>
<expression label="F_2" exp="(c, 0)" type="point" />
<element type="point" label="F_2">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="5" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large %n%v$"/>
</element>
<command name="Circle">
<input a0="F_1" a1="(2 * a)"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="63"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-39" A3="0" A4="5" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="d"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<auxiliary val="true"/>
<coords x="2.2074745250823504" y="3.4716438426614205" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large %n$"/>
</element>
<command name="Line">
<input a0="D" a1="F_1"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="3.4716438426614205" y="-7.20747452508235" z="17.358219213307102"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="63"/>
<eqnStyle style="implicit"/>
</element>
<command name="Segment">
<input a0="D" a1="F_2"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="3.4716438426614205" y="2.7925254749176496" z="-17.358219213307102"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="63"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="LineBisector">
<input a0="g"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-2.7925254749176496" y="3.4716438426614205" z="4.037372625411749"/>
<lineStyle thickness="5" type="30" typeHidden="1" opacity="63"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="f" a1="h"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-22" y="0"/>
<labelMode val="3"/>
<auxiliary val="true"/>
<coords x="89.36081519736479" y="62.48958916790556" z="8.074745250823497"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large %n$"/>
</element>
<command name="Locus">
<input a0="P" a1="D"/>
<output a0="双曲线"/>
</command>
<element type="locus" label="双曲线">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="b" exp="sqrt(c^(2) - a^(2))" />
<element type="numeric" label="b">
<value val="3"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<expression label="text1" exp=""\frac{x^2}{" + (LaTeX[a]) + "^2}-\frac{y^2}{" + (LaTeX[b]) + "^2}=1"" />
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="102" y="232"/>
</element>
<expression label="text2" exp=""b=\sqrt{c^2-a^2}\\
=\sqrt{" + (LaTeX[c]) + "^2-" + (LaTeX[a]) + "^2}\\
=" + (LaTeX[sqrt(c^(2) - a^(2))]) + """ />
<element type="text" label="text2">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="104" y="133"/>
</element>
<command name="Segment">
<input a0="P" a1="F_2"/>
<output a0="PF_2"/>
</command>
<element type="segment" label="PF_2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="51" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="7.738892959072932" y="-6.066703954313777" z="-38.69446479536466"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="63"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P" a1="F_1"/>
<output a0="PF_1"/>
</command>
<element type="segment" label="PF_1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="51" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="7.738892959072932" y="-16.066703954313777" z="38.69446479536466"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="63"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text4" exp=""2a=" + (LaTeX[(2 * a)]) + """ />
<element type="text" label="text4">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="106" y="348"/>
</element>
<expression label="text3" exp=""||PF_1|-|PF_2||\\
=|" + (LaTeX[PF_1]) + "-" + (LaTeX[PF_2]) + "|\\
=" + (LaTeX[abs(PF_1 - PF_2)]) + """ />
<element type="text" label="text3">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="102" y="381"/>
</element>
</construction>
</geogebra>