如图所示,已知抛物线C:x²=4y,过点M(0,2)任作一直线与C相交于A,B两点,过点B作y轴的平行线与直线AO相交于点D(O为坐标原点).⑴证明:动点D在定直线上⑵作C的任意一条切线L,(不含x轴),与直线y=2相
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![如图所示,已知抛物线C:x²=4y,过点M(0,2)任作一直线与C相交于A,B两点,过点B作y轴的平行线与直线AO相交于点D(O为坐标原点).⑴证明:动点D在定直线上⑵作C的任意一条切线L,(不含x轴),与直线y=2相](/uploads/image/z/7258840-16-0.jpg?t=%E5%A6%82%E5%9B%BE%E6%89%80%E7%A4%BA%2C%E5%B7%B2%E7%9F%A5%E6%8A%9B%E7%89%A9%E7%BA%BFC%3Ax%26%23178%3B%3D4y%2C%E8%BF%87%E7%82%B9M%280%2C2%29%E4%BB%BB%E4%BD%9C%E4%B8%80%E7%9B%B4%E7%BA%BF%E4%B8%8EC%E7%9B%B8%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E8%BF%87%E7%82%B9B%E4%BD%9Cy%E8%BD%B4%E7%9A%84%E5%B9%B3%E8%A1%8C%E7%BA%BF%E4%B8%8E%E7%9B%B4%E7%BA%BFAO%E7%9B%B8%E4%BA%A4%E4%BA%8E%E7%82%B9D%28O%E4%B8%BA%E5%9D%90%E6%A0%87%E5%8E%9F%E7%82%B9%29.%E2%91%B4%E8%AF%81%E6%98%8E%3A%E5%8A%A8%E7%82%B9D%E5%9C%A8%E5%AE%9A%E7%9B%B4%E7%BA%BF%E4%B8%8A%E2%91%B5%E4%BD%9CC%E7%9A%84%E4%BB%BB%E6%84%8F%E4%B8%80%E6%9D%A1%E5%88%87%E7%BA%BFL%2C%28%E4%B8%8D%E5%90%ABx%E8%BD%B4%29%2C%E4%B8%8E%E7%9B%B4%E7%BA%BFy%3D2%E7%9B%B8)
如图所示,已知抛物线C:x²=4y,过点M(0,2)任作一直线与C相交于A,B两点,过点B作y轴的平行线与直线AO相交于点D(O为坐标原点).⑴证明:动点D在定直线上⑵作C的任意一条切线L,(不含x轴),与直线y=2相
如图所示,已知抛物线C:x²=4y,过点M(0,2)任作一直线与C相交于A,B两点,过点B作y轴的平行线与直线AO相交于点D(O为坐标原点).
⑴证明:动点D在定直线上
⑵作C的任意一条切线L,(不含x轴),与直线y=2相交于点N1,与⑴中的定直线相交于点N2,证明:lMN2l²-lMN1l²为定值,并求此定值
如图所示,已知抛物线C:x²=4y,过点M(0,2)任作一直线与C相交于A,B两点,过点B作y轴的平行线与直线AO相交于点D(O为坐标原点).⑴证明:动点D在定直线上⑵作C的任意一条切线L,(不含x轴),与直线y=2相
(1)
令A(a,a²/4),B(b,b²/4)
AB:(y - b²/4)/(a²/4 - b²/4) = (x - b)/(a - b)
过M(0,2),从上式可得b = -8/a
BD:x = -8/a
OA:y = ax/4,D(-8/a,-2)
D在y = -2上
(2)
取抛物线上的点P(p,p²/4)
y = x²/4,y' = x/2
过P的切线斜率:k = p/2
切线:y - p²/4 = (p/2)(x - p)
令y = 2,x = c/2 + 4/c,N1(c/2 + 4/c,2)
取y = -2,x = c/2 - 4/c,N2(c/2 - 4/c,-2)
|MN1|² = (c/2 + 4/c)² = c²/4 + 4 + 16/c²
|MN2|² = (c/2 - 4/c)² + (2 + 2)² = c²/4 - 4 + 16/c² + 16
|MN2|² - |MN1|² = 8