已知直线y=x-2与抛物线y^2=ax相交于AB两点,且OA垂直OB,求实数A

代入
(x-2)^2=ax^2
x^2-(4+a)x+4=0
x1+x2=a+4,x1x2=4
y=x-2
所以y1y2=(x1-2)(x2-2)=x1x2-2(x1+x2)+4=-2a
OA斜率y1/x1,OB斜率y2/x2
垂直(y1/x1)(y2/x2)=-1
y1y2=-x1x2
-2a=-4
a=2

(x-2)^2=ax
x^2-4x+4=ax
x^2-x(4+a)+4=0
x1+x2=a+4
x1*x2=4
三角形OAB是直角三角形,OA^2+OB^2=AB^2
(x1^2+y1^2)+(x2^2+y2^2)=(x1-x2)^2+(y1-y2)^2
-2x1*x2-2y1*y2=0
-8-2(x1-2)(x2-2)=0
2x1*x2-4(x1+x2)+8+8=0
8-4(a+4)+16=0
a=2