Line Passing Through a Fixed Point on Conic Sections
2021-01-07 00:40
This GeoGebra example demonstrates the classic fixed-point line problem in coordinate geometry. It constructs a conic section with foci F₁ and F₂, using commands such as Point, Segment, OrthogonalLine, and Intersect to draw lines, perpendiculars, and intersection points, with angles α and β annotated for geometric analysis. Interactive buttons enable dynamic exploration of the property that a fami
In collections
Conic Sections - Interactive GeoGebra Book
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="w" id="5121e619-a4b5-4aba-91f7-0fc9b26bc661" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1334" height="536" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.08170914542728636" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="333,42,699,643" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" 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="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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 | 100013 100015" visible="false" inframe="false" stylebar="false" location="1,1" size="800" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1215" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="109" tab="ALGEBRA" 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="712" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" 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" />
</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 | | 100015 100014 100014 100013 100012 100010 100009 100010 100002 100003 100004 100005" position="1" help="false" />
<input show="false" cmd="true" top="true" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1215" height="534"/>
<coordSystem xZero="599.8157033630098" yZero="267.0744502606394" scale="109.05194665671294" yscale="109.05192737366438"/>
<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"/>
<labelStyle axes="0" serif="true"/>
<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="1"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<startAnimation val="true"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.119946536715575" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{%n}$"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="1.2386250615984928" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{%n}$"/>
</element>
<expression label="eq1" exp="x^(2) / x(A)^(2) + y^(2) / y(B)^(2) = 1" type="conic" />
<element type="conic" label="eq1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="0.4450206652584395" A1="1.3036177633945487" A2="-2" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<expression label="F_2" exp="(sqrt(x(A)^(2) - y(B)^(2)), 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="1.7204596116481465" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{F_2}$"/>
</element>
<expression label="F_1" exp="((-sqrt(x(A)^(2) - y(B)^(2))), 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="-1.7204596116481465" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{F_1}$"/>
</element>
<expression label="C" exp="(0, sqrt(y(B)^(2) - x(A)^(2)))" type="point" />
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="0" y="NaN" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{F_2}$"/>
</element>
<expression label="D" exp="(0, (-sqrt(y(B)^(2) - x(A)^(2))))" type="point" />
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="0" y="NaN" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{F_1}$"/>
</element>
<command name="Point">
<input a0="eq1"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-28" y="2"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.5275831198704498" y="1.1996552723914762" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{P}$"/>
</element>
<expression label="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{O}$"/>
</element>
<expression label="m" exp="abs(x(A))" />
<element type="numeric" label="m">
<value val="2.119946536715575"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="n" exp="abs(y(B))" />
<element type="numeric" label="n">
<value val="1.2386250615984928"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Point">
<input a0="eq1"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="34"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="true"/>
<coords x="0.33926019077719344" y="-1.2226613616280932" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{%n}$"/>
</element>
<command name="Segment">
<input a0="P" a1="E"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.4223166340195696" y="0.8668433106476432" z="0.23806021913440356"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="f"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8668433106476432" y="2.4223166340195696" z="-3.3632768196734637"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="eq1" a1="g" a2="2"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-28" y="-1"/>
<labelMode val="3"/>
<coords x="-1.5892939045362615" y="0.8197144840549734" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{%n}$"/>
</element>
<command name="Segment">
<input a0="F" a1="P"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3799407883365028" y="1.0617107846658116" z="-1.4741372870558425"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="E"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.0423758456830665" y="1.928554095313455" z="1.6650717571042155"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="F" a1="P" a2="E"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<condition showObject="α ≟ π / 2"/>
<objColor r="32" g="32" b="32" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="E" a1="P" a2="F"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="4.71238898038469"/>
<show object="true" label="false" ev="4"/>
<condition showObject="β ≟ π / 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="Line[B, P]"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5275831198704498" y="0.03896978920701666" z="0.23159363528605542"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="eq1" a1="j" a2="2"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.34872651720142356" y="-1.2217518320781595" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="Line[A, P]"/>
<output a0="k"/>
</command>
<element type="line" label="k">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.647529656586025" y="-1.1996552723914762" z="2.8359647287478627"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="eq1" a1="k" a2="2"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.474441189501714" y="-0.8899740383098125" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="Line[A, H]" a1="Line[B, G]"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-2" y="42"/>
<labelMode val="3"/>
<coords x="-2.210506286880889" y="-5.026403275302547" z="8.533190961720512"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{Q}$"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="135"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="a=!a
SetTrace(i,a)
ZoomIn(1)
If(a==true,SetCaption(button1,"取消跟踪"),SetCaption(button1,"跟踪"))"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="跟踪"/>
</element>
<element type="boolean" label="a">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="10" y="10"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="236"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$\large{显示＼隐藏定点}$"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="187"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="c=!c
If(c==true,StartAnimation(E,true),StartAnimation(E,false))
If(c==true,SetCaption(button2,"暂停动画“Ｅ”"),SetCaption(button2,"启动动画“Ｅ”"))"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="暂停动画“Ｅ”"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="80" y="45"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text1" exp=""注：拖动点“A”、“B”、“P”,改变位置.""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="191" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="10" y="494"/>
</element>
<expression label="text3" exp=""直线过定点问题(1)""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="0" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="0"/>
<absoluteScreenLocation x="12" y="14"/>
</element>
<expression label="text2" exp=""椭圆方程：\frac{x^2}{m^2}+\frac{y^2}{n^2}=1,即\frac{x^2}{" + (LaTeX[m]) + "^2}+\frac{y^2}{" + (LaTeX[n]) + "^2}=1"" />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="384" y="11"/>
</element>
</construction>
</geogebra>