作业帮已知直线Ax+By+C=0（其中A*A+B*B=C*C,C≠0)与圆x*x+y*y=4交于M,N,O是坐标原点,则OM的
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已知直线Ax+By+C=0（其中A*A+B*B=C*C,C≠0)与圆x*x+y*y=4交于M,N,O是坐标原点,则OM的向量乘ON的向量等于
L:Ax+By+C =0 (1)
A^2 +B^2 = C^2
C:x^2 + y^ 2= 4 (2)
let M(x1,y1) ,N(x2,y2)
sub (1) into (2)
x^2 + [(C-Ax)/B]^2 = 4
(1+ (A/B)^2)x^2 - (2AC/B)x +(C/B)^2-4 =0
x1x2 = [(C/B)^2-4] / (1+ (A/B)^2)
= (C^2 -4B^2)/(B^2+ A^2)
= (C^2 -4B^2)/(C^2)
= 1- 4(B/C)^2
Similary ,we have
((C-By)/A)^2 + y^2 = 4
(1+ (B/A)^2)y^2 - (2BC/A)y +(C/A)^2-4 =0
y1y2 = [(C/A)^2-4]/ (1+ (B/A)^2)
= (C^2-4A^2)/(A^2+B^2)
= (C^2-4A^2)/(C^2)
= 1- 4(A/C)^2
OM.ON
= x1x2+y1y2
= 1- 4(B/C)^2 + 1- 4(A/C)^2
= 2 - 4(B^2+A^2)/C^2
= 2-4
=-2
A^2 +B^2 = C^2
C:x^2 + y^ 2= 4 (2)
let M(x1,y1) ,N(x2,y2)
sub (1) into (2)
x^2 + [(C-Ax)/B]^2 = 4
(1+ (A/B)^2)x^2 - (2AC/B)x +(C/B)^2-4 =0
x1x2 = [(C/B)^2-4] / (1+ (A/B)^2)
= (C^2 -4B^2)/(B^2+ A^2)
= (C^2 -4B^2)/(C^2)
= 1- 4(B/C)^2
Similary ,we have
((C-By)/A)^2 + y^2 = 4
(1+ (B/A)^2)y^2 - (2BC/A)y +(C/A)^2-4 =0
y1y2 = [(C/A)^2-4]/ (1+ (B/A)^2)
= (C^2-4A^2)/(A^2+B^2)
= (C^2-4A^2)/(C^2)
= 1- 4(A/C)^2
OM.ON
= x1x2+y1y2
= 1- 4(B/C)^2 + 1- 4(A/C)^2
= 2 - 4(B^2+A^2)/C^2
= 2-4
=-2
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