湖北州二中 操厚亮
摘 文研究圆锥曲线中定点点定分点弦存问题出圆锥曲线中定弦存较般判定定理
关键词 圆锥曲线 定点 中点弦 定弦
The Existence Theroem of Fixed proportion
Nypothenuse in Conical Curye
Cao Houliang
(Class 9702Department of Mathematics
Hubei Normal University)
Abstract
In this paperwe carry out a research into the existence problem of acertain hypothenuse which passes through a fixed point and has it as a fixedproportion pointin conical curve give out several common theorems to judge the existence of fixedproportion hupothenuse in conical curve
Key Wordconical curvefixed pointcenterpoint hypothenusefixedproportion hypothenuse
首先出定义:
定义 设P点定点T圆锥曲线AB弦AB直线P点P点分成线段代数长(定值)AB便做T定弦时定弦中点弦
文研究定弦存定理
定理 椭圆存P()(x02+ y02≠0)分点定定弦充条件:
(1)>0时()≤b2x02+a2y02<a2b2
(2)0时b2x02+a2y02a2b2(Ⅰ)
(3)<0时(≠1)<b2x02+a2y02≤()
证明:设A(xy)
B()
b2x2+a2y2a2b2
b2[(1+)x0x]2+a2[(1+)y0y]2a2b22(*)
两式相减
b2(1+)2x022b2(1+)x0+a2(1+)2y022a2(1+)y0ya2b2(21)0(*)
①y0≠0时
y·
代入化简:
()假设弦AB存述方程实根△≥0化简整理:≤0解等式
:
(1)>0时()≤b2x02+a2y02<a2b2
(2)0时b2x02+a2y02a2b2
(3)<0时(≠1)<b2x02+a2y02≤()
②0时时P点(x00)
(**):x
≥
≥
(1)>0时()≤x02<a2
(2)0时 x02a2
(3)<0时<x02≤()
结(Ⅰ)式中取情形否零(Ⅰ)式总成立
()反(Ⅰ)式成立推导程逆P(x0y0)分点定定弦必存
x00时y00时P椭圆中心时相应弦中点弦改变改变中点弦唯P点椭圆中心
综述知定理定成立
定理二 抛物线y22px(p>0)存(x0y0)分点定定弦充条件:
(1)≠0(≠1)时()<0
(2)0时 (Ⅱ)
证明:设A(xy)
B()
(* *)
两式相减:
(* *)
①y0≠0时y
代入y22px
()设弦AB存xR∴方程实根∴△≥0化简:
(1)≠0(≠1)(y022px0)<0
(2)0时y022px0
②y00时时P点(x00)
(* *)x0xy22pyy22px0≥0≠0时x0>0
0时x00
结(Ⅱ)式中取y00时情形y0否零(Ⅱ)式总成立
反(Ⅱ)式成立推导程逆P(x0y0)分点定定弦必存
定理三 双曲线存P()(x02+y02≠0)分点定定弦充条件:
(1)>0时b2x02a2y02≤()b2x02a2y02<a2b2
(2)0时b2x02a2y02a2b2 (Ⅲ)
(3)<0时b2x02a2y02≥()b2x02a2y02<a2b2
证明前面类似
证明定弦存定理中点弦存定理证明相应定理需述定理中改1述推:
推 椭圆b2x02+a2y02 a2b2存P(x0y0)(x02+y02≠0)中点中点弦充条件:
b2x02+a2y02<a2b2(Ⅳ)
推二 抛物线y22px存P(x0y0)中点中点弦充条件:
y02<2px0(Ⅴ)
推三 双曲线b2x2a2y2a2b2 存P(x0y0)(x02+y02≠0)中点中点弦充条件
b2x02a2y02>a2b2b2x02a2y02<0 (Ⅵ)
推四 圆x2+y2R2存P(x0y0)(x02+y02≠0)中点中点弦充条件:
x02+y02<R2(Ⅶ)
面举例定弦存定更换应举例:
例1 椭圆4x2+9y236存P(x0y0)分点2定定弦求P点存范围
解:定理1知<0(≠1)时椭圆b2x2+a2y2a2b2存P(x0y0)分点定定弦充条件<b2x02+a2y02≤()a29b242代入36<4x02+9y02≤324P点存范围椭圆4x2+9y2364x2+9y2324夹区域(含4x2+9y2324)
例2 P(x0y0)区域双曲线x24y24存P(x0y0)分点2定定弦?
解:定理三知<0(≠1)时双曲线存P()分点定定弦充条件b2x02a2y02≥()b2x02a2y02<a2b2a24b21 2代入x024y02≥36x024y02<4P点区域x024y02<36x024y02≥4双曲线x24y236x24y24夹区域(含双曲线x24y24)
例3 点P(12)作椭圆x2+4y24弦ABP点分AB成线段求值
解:∵P(12)椭圆x2+4y24外点∴P外分点<0定理知该椭圆存P(12)分点定定弦充条件4()2≥17解等式:
≤<11<≤
∴值值
例4 点A(11)直线L双曲线交P1 P2两点求线段P1 P2中点P轨迹方程
解:设P1(x1y1)P2(x2y2)两式相减化简
设P点坐标x式化2xyk0∴kP1P2斜率∴直线L方程
化简整理
推3知双曲线存P(x0y0)(x02+y02≠0)中点中点弦充条件:
b2x02+a2y02<0b2x02+a2y02>a2 b2
求P点轨迹方程
2x2y22x+y0
<0>2
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