由动点P引圆x2+y2=10的两条切线PA,PB...

1) 若k1+k2+k1×k2=-1,求动点P的轨迹方程
设点P为(a,b),
直线为y-b=k(x-a)
代入圆方程
x²+(kx-ak+b)²=10
(1+k²)x²-2kx(ak-b)+(ak-b)²-10=0
因直线与圆相切则方程仅有一实根
则4k²(ak-b)²=4(1+k²)[(ak-b)²-10]
a²k^4-2abk³+b²k²=a²k^4-2k³ab+k²(b²-10)+a²k²-2abk+b²-10
(a²-10)k²-2abk+b²-10=0
则k1+k2=2ab/(a²-10),k1*k2=(b²-10)/(a²-10)
因k1+k2+k1×k2=-1,
则2ab/(a²-10)+(b²-10)/(a²-10)=-1
2ab+a²-10+b²-10=0
(a+b)²=20
P点轨迹为x+y=±2√5两直线,除点(±√5,±√5)两个点以为.
2) 若点P在直线x+y=m上,且AP⊥BP,求实数m的取值范围
已证k1*k2=(b²-10)/(a²-10)
AP⊥BP
则k1*k2=-1
则(b²-10)/(a²-10)=-1
a²+b²=20
P点轨迹为x²+y²=20
有P在直线x+y=m上
则(m-y)²+y²-20=0
y²-my+m²/2-10=0
则m²>=4(m²/2-10)
m²