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
来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/24 00:33:43
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