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高二数学抛物线及其标准方程2_图文

复习回顾

1.抛物线的定义是什么?

平面内,到定点的距离和到定直 线的距离相等的点的轨迹叫抛物线
2.抛物线的标准方程有哪几种形式?

(一)、回顾
图 形
y
l O F x l

方 程

焦 点

准 线

y2 = 2px (p>0) y2 = -2px
x F x l l F x

p F ( ,0 ) 2 p F ( ? ,0) 2 p F (0, ) 2 p F (0,? ) 2

y
F

p x?? 2
p x? 2 p y?? 2 p y? 2

O

(p>0) x2 = 2py (p>0) x2 = -2py (p>0)

y
O

y
O

y

例题:
解:如图,设点M的坐标为(x,y), 由已知条件可知:点M与点F的距离 等于它到直线x+4=0的距离,根据 抛物线的定义,点M的轨迹是以 F(4,0)为焦点的抛物线。

例1:点M与点F(4,0)的距离比它到直线l:x+5=0的 y 距离小1,求点M的轨迹方程。 M
O F X

p ? 4, 2

?p ?8

L L’

∵焦点在x轴的正半轴上,∴点M的轨迹方程为:

y2=16x

演示

例题:
例2:已知抛物线的焦点在x轴上,抛物线上的 点M(-3,m)到焦点的距离等于5,求抛物线的标 准方程和m的值。 解 因为焦点在x轴上且过M点的

所以设标准方程为 y2=-2px(p>0),

p 由抛物线的定义知 -(-3)=5 2
即p=4.

所以所求抛物线标准方程为y2=-8x 又点M(-3,m)在抛物线上,于是m2=24, 得: m ? ?2 6

练习:
1、动点P到点A(0,2)的距离比到直线l:y=-4的 X2=8y 距离小2,则动点P的轨迹方程为________

2、已知抛物线方程为标准方程,焦点在y轴上 抛物线上一点M(a,-4)到焦点F的距离是5,则 抛物线方程为_______ X2=- 4y ,a的值等于____ ±4

拓展:
斜率为1的直线经过抛物线y2=4x的焦点与 y 抛物线相交于两点A、B,求线段AB的长。A‘ 解: 由抛物线方程知,焦点F(1,0) 所以直线方程为y=x-1 O 联立 y=x-1 消去y得x2-6x+1=0 B’ B
A F

X

y2=4x

? x1 ? x2 ? 6

设A(x1,y1),B(x2,y2),由抛物 线的定义可知: |AF|等于点A到准线x=-1的距离|AA’|, 而|AA’|=x1+1 即|AF|=|AA’|=x1+1,同理|BF|=|BB’|=x2+1 ∴|AB|=|AF|+|BF|=x1+x2+2=8

小结:
抛物线的定义反映了抛物线的本
质,灵活应用定义往往可以化繁为简、 化难为易,且思路清晰,解法简捷, 巧妙解法常常来源于对定义的恰当运 用,同学们要好好体会。

作业:
P119 T5、6


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