关于抛物线的题目1.已知抛物线的顶点在坐标原点,焦点在Y轴上.抛物线上的点(M,-2)到焦点的距离等于4,则M=?2.已知抛物线Y^2=2PX(P大于0)的焦点F,P1(x1.y1),P2(x2,y2),P3(x3,y3)在抛物线上,且2*X2=X1+X3,则:(答案中都有绝对值的)AFP1+FP2=FP3B.FP1^2+FP2^2=FP3C.2FP2=FP1+FP3D.FP2^2=FP1*FP33.已知抛物线Y^2=2px(P大于0),焦点为F,一直线L与抛物线交与A.B俩点,IAFI+IBFI=8.且AB的垂直平分线恒过定点S(6,0),求抛物线方程

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关于抛物线的题目
1.已知抛物线的顶点在坐标原点,焦点在Y轴上.抛物线上的点(M,-2)到焦点的距离等于4,则M=?
2.已知抛物线Y^2=2PX(P大于0)的焦点F,P1(x1.y1),P2(x2,y2),P3(x3,y3)在抛物线上,且2*X2=X1+X3,则:(答案中都有绝对值的)
AFP1+FP2=FP3
B.FP1^2+FP2^2=FP3
C.2FP2=FP1+FP3
D.FP2^2=FP1*FP3
3.已知抛物线Y^2=2px(P大于0),焦点为F,一直线L与抛物线交与A.B俩点,IAFI+IBFI=8.且AB的垂直平分线恒过定点S(6,0),求抛物线方程

1.点(M,-2)到焦点的距离等于4
-->(M,-2)到准线距离为4
所以准线为y=2
所以p/2=-2 -->p=-4 2p=-8
x^2=2py -->x^2=-8y
所以y=-2 x=M=±4
2.FP1=x1+p/2 FP2=x2+p/2 FP3=x3+p/2
由x1+x3=2x2 -->2FP2=FP1+FP3
(用抛物线定义)
所以为C
3.① 焦点在x轴上,可设抛物线方程为:y² = 2px.可以判断焦点在(p/2,0)点.
② 设A点坐标(x1,y1),B点坐标(x2,y2),设AB斜率是k,线段AB的垂直平分线斜率是k'
则:kk' = -1,所以:
(y1-y2)/(x1-x2) * [(y1+y2)/2 - 0 ]/[(x1+x2)/2 - 6] = -1
(y1² - y2²) / [x1² - x2² -12(x1 - x2)] = -1
代入y1²=2px1,y2²=2px2,化简:
2p/(x1 + x2 - 12) = -1
x1 + x2 = 12 - 2p ---

AF²=(x1 - p/2)² + y1² = (x1 - p/2)² + 2px1 = (x1 + p/2)²
AF = x1 + p/2
同理:
BF = x2 + p/2
AF + BF = x1 + x2 + p ---
link:
12 - 2p + p = 8
p=4
综上:
抛物线方程:
y² = 8x