斜率为k的直线经过抛物线y^2=2px的焦点F,并与抛物线相交于两点A(x1,y1),B(x2,y2)证明:(1)y1*y2=-p^2(2)x1*x2=(p^2)/4
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![斜率为k的直线经过抛物线y^2=2px的焦点F,并与抛物线相交于两点A(x1,y1),B(x2,y2)证明:(1)y1*y2=-p^2(2)x1*x2=(p^2)/4](/uploads/image/z/8657211-3-1.jpg?t=%E6%96%9C%E7%8E%87%E4%B8%BAk%E7%9A%84%E7%9B%B4%E7%BA%BF%E7%BB%8F%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D2px%E7%9A%84%E7%84%A6%E7%82%B9F%2C%E5%B9%B6%E4%B8%8E%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9B%B8%E4%BA%A4%E4%BA%8E%E4%B8%A4%E7%82%B9A%28x1%2Cy1%29%2CB%28x2%2Cy2%29%E8%AF%81%E6%98%8E%3A%EF%BC%881%EF%BC%89y1%2Ay2%3D-p%5E2%EF%BC%882%EF%BC%89x1%2Ax2%3D%28p%5E2%29%2F4)
斜率为k的直线经过抛物线y^2=2px的焦点F,并与抛物线相交于两点A(x1,y1),B(x2,y2)证明:(1)y1*y2=-p^2(2)x1*x2=(p^2)/4
斜率为k的直线经过抛物线y^2=2px的焦点F,并与抛物线相交于两点A(x1,y1),B(x2,y2)
证明:(1)y1*y2=-p^2
(2)x1*x2=(p^2)/4
斜率为k的直线经过抛物线y^2=2px的焦点F,并与抛物线相交于两点A(x1,y1),B(x2,y2)证明:(1)y1*y2=-p^2(2)x1*x2=(p^2)/4
弦AB斜率
k=(y1-y2)/(x1-x2)
=(y1-y2)/[(y1^2/2p)-(y2^2/2p)]
=2p/(y1+y2) (1)
而A、F、B三点共线,故
k=(y1-0)/(x1-p/2) (2)
由(1)、(2)得
y1/(x1-p/2)=2p/(y1+y2)
--->y1y2+y1^2=2px1-p^2
而y1^2=2px1
故y1y2=-p^2
又x1x2=(y1^2/2p)×(y2^2/2p)
=(y1y2)^2/(4p^2)
=(-p^2)^2/(4p^2)
故x1x2=(p^2)/4
求该抛物线的方程 |AB|=x1+p/2+x2+p/2=x1+x2+p (x1+x2)(x1-x2)^2=9 y=k(x-p/2) k^2(x^2-px+p^2/4)=2px k^2x^
易知k不为0,焦点为F(p/2,0),设直线方程为y=k(x-p/2),则
x= y/k +p/2
代入 y²=2px,得 y²=2py/k+p²
y²-2py/k-p²=0
所以 y1y2=-p²
x1x2=[y1²/(2p)][y2²/(2p)]=(y1y2)²/...
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易知k不为0,焦点为F(p/2,0),设直线方程为y=k(x-p/2),则
x= y/k +p/2
代入 y²=2px,得 y²=2py/k+p²
y²-2py/k-p²=0
所以 y1y2=-p²
x1x2=[y1²/(2p)][y2²/(2p)]=(y1y2)²/(4p²)=p²/4
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解:k=(y1-y2)/(x1-x2)
=(y1-y2)/[(y1^2/2p)-(y2^2/2p)]
=2p/(y1+y2) (1)
而A、F、B三点共线,故
k=(y1-0)/(x1-p/2) (2)
由(1)、(2)得
y1/(x1-p/2)=2p/(y1+y2)
--->y1y2+y1^2=2px1-p^2
然后y1^2=2px...
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解:k=(y1-y2)/(x1-x2)
=(y1-y2)/[(y1^2/2p)-(y2^2/2p)]
=2p/(y1+y2) (1)
而A、F、B三点共线,故
k=(y1-0)/(x1-p/2) (2)
由(1)、(2)得
y1/(x1-p/2)=2p/(y1+y2)
--->y1y2+y1^2=2px1-p^2
然后y1^2=2px1
因此y1y2=-p^2
又x1x2=(y1^2/2p)×(y2^2/2p)
=(y1y2)^2/(4p^2)
因此=(-p^2)^2/(4p^2)x1x2=(p^2)/4
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