已知两点M(-1,0),N(1,0),且点P(x,y)使得向量MP×向量MN,向量PM×向量PN,向量NM×向量NP成公差小于零的等差数列.(1)求证 x²+y²=3(x>0)(2)若点P的坐标为(x0,y0),记向量PM与向量PN
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![已知两点M(-1,0),N(1,0),且点P(x,y)使得向量MP×向量MN,向量PM×向量PN,向量NM×向量NP成公差小于零的等差数列.(1)求证 x²+y²=3(x>0)(2)若点P的坐标为(x0,y0),记向量PM与向量PN](/uploads/image/z/5543003-11-3.jpg?t=%E5%B7%B2%E7%9F%A5%E4%B8%A4%E7%82%B9M%EF%BC%88-1%2C0%EF%BC%89%2CN%EF%BC%881%2C0%EF%BC%89%2C%E4%B8%94%E7%82%B9P%EF%BC%88x%2Cy%EF%BC%89%E4%BD%BF%E5%BE%97%E5%90%91%E9%87%8FMP%C3%97%E5%90%91%E9%87%8FMN%2C%E5%90%91%E9%87%8FPM%C3%97%E5%90%91%E9%87%8FPN%2C%E5%90%91%E9%87%8FNM%C3%97%E5%90%91%E9%87%8FNP%E6%88%90%E5%85%AC%E5%B7%AE%E5%B0%8F%E4%BA%8E%E9%9B%B6%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.%EF%BC%881%EF%BC%89%E6%B1%82%E8%AF%81+x%26%23178%3B%2By%26%23178%3B%3D3%EF%BC%88x%EF%BC%9E0%EF%BC%89%EF%BC%882%EF%BC%89%E8%8B%A5%E7%82%B9P%E7%9A%84%E5%9D%90%E6%A0%87%E4%B8%BA%EF%BC%88x0%2Cy0%EF%BC%89%2C%E8%AE%B0%E5%90%91%E9%87%8FPM%E4%B8%8E%E5%90%91%E9%87%8FPN)
已知两点M(-1,0),N(1,0),且点P(x,y)使得向量MP×向量MN,向量PM×向量PN,向量NM×向量NP成公差小于零的等差数列.(1)求证 x²+y²=3(x>0)(2)若点P的坐标为(x0,y0),记向量PM与向量PN
已知两点M(-1,0),N(1,0),且点P(x,y)使得向量MP×向量MN,向量PM×向量PN,向量NM×向量NP成公差
小于零的等差数列.(1)求证 x²+y²=3(x>0)
(2)若点P的坐标为(x0,y0),记向量PM与向量PN的夹角为a,求tana
已知两点M(-1,0),N(1,0),且点P(x,y)使得向量MP×向量MN,向量PM×向量PN,向量NM×向量NP成公差小于零的等差数列.(1)求证 x²+y²=3(x>0)(2)若点P的坐标为(x0,y0),记向量PM与向量PN
向量MP: (x-(-1),y-0) = (x+1,y)
向量PN: (1-x,0-y) = (1-x,-y)
向量NM: (-1-1,0-0) = (-2,0)
向量MN: (1-(-1),0-0) = (2,0)
向量NP: (x-1,y-0) = (x-1,y)
向量PM: (-1-x,0-y) = (-x-1,-y)
等差数列公差:
向量NM*向量NP- 向量PM*向量PN=向量PM*向量PN-向量MP*向量MN
(-2,0)(x-1,y) - (-x-1,-y) (1-x,-y) = (-x-1,-y) (1-x,-y) - (x+1,y)(2,0)
(-2*(x-1)+0*y) - ((-x-1)*(1-x)+(-y)*(-y)) = ((-x-1)*(1-x)+(-y)*(-y)) - ((x+1)*2+0*y)
simplify:
-x (x+2)-y^2+3 = (x-2) x+y^2-3
-2 x^2-2 y^2+6 = 0
it is a circle:
x^2+y^2 = 3 证毕(1)
(2) see Figure
k2=y/(x-1), k1=y/(x+1)
tan(a)=(k2-k1)/(1-k2*k1)=(2 y)/(x^2-y^2-1)
=2*y/((3-y^2)-y^2-1)=1/y