点到双曲线的切线方程
2020-01-08 13:26
本范例通过点到直线的距离概念，直观演示如何求解和理解点到双曲线的切线方程。利用GeoGebra的切线和极线等工具，动态展示点到双曲线的切线构造过程，帮助学习者深入掌握圆锥曲线中切线相关的解析几何性质与几何作图方法。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.570.0" app="classic" platform="d" id="0d22b97f-b028-46dd-83c4-0b63e7536142" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1938" height="1048" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.1259029927760578" orientation="1" />
<pane location="1" divider="0.5159958720330238" 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="1" visible="true" inframe="false" stylebar="false" location="1,3" size="1000" 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="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="4097" visible="true" inframe="false" stylebar="true" location="1,1" size="655" window="610,240,700,550" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="244" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" 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 , 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 | 100001 | 44 | 100002 | 100003 | 100004 | 100015 100005" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1000" height="858"/>
<coordSystem xZero="578.4918696142106" yZero="399.10085583590205" scale="65.94693996176181" yscale="65.94696291569855"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="2" 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"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="4"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="eq1" exp="y^(2) - x^(2) = 1" type="conic"/>
<element type="conic" label="eq1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="-0.0" y0="1.0" z0="1.0" x1="-1.0" y1="-0.0" z1="1.0"/>
<matrix A0="-1.0" A1="1.0" A2="-1.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="implicit"/>
</element>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-17" y="-10"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-5.0" y="-2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Tangent">
<input a0="A" a1="eq1"/>
<output a0="h" a1="i"/>
</command>
<element type="line" label="h">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.212003752338912" y="5.684801500935565" z="-4.690415759823431"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<element type="line" label="i">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.9784724381372789" y="4.791388975254912" z="4.690415759823428"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="Tangent[A, eq1]"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.1883517156472787" y="0.15364223182150594" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="c" exp="sqrt((x(A)^(2) + y(A)^(2)) / (x(P)^(2) + y(P)^(2)))" />
<element type="numeric" label="c">
<value val="4.494218486158932"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="line" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="5.0" y="-2.0" z="-1.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="pdp">
<input a0="P" a1="f"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.7830154556259448" y="-10.042461360935139" z="29.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.49993400288823486" y="1.249835007220587" z="0.4020699901246265"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="pdp">
<input a0="A" a1="f"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-35.0" y="-102.0" z="29.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.5172413793103448" y="3.793103448275862" z="15.172413793103447"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="a" exp="g^(2)" />
<element type="numeric" label="a">
<value val="1.8120215525179393"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Polar">
<input a0="P" a1="eq1"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.1883517156472787" y="0.15364223182150594" z="-1.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="pdp">
<input a0="P" a1="j"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.1322474079853357" y="0.5875831442571471" z="1.435785735480924"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="D"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.25559929338443504" y="1.97694250604546" z="-0.6074837176232509"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="b" exp="g^(2) - (l * k / sqrt((x(A)^(2) + y(A)^(2)) / (x(P)^(2) + y(P)^(2))))" />
<element type="numeric" label="b">
<value val="1.1102230246251565E-15"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="image" label="pic1">
<file name="b5a49d1a736d6530742b3763d1e5b2b7/transferImage.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="-5" y="382"/> <auxiliary val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="false"/>
</element>
</construction>
</geogebra>