C:(x-1)^2＋(y-2)^2=4
A=(1,4)
B=(2,4)
1.過A求圓的切線(check A on C)
方法一.點到直線距離=r ( d(O,L)=r )
check A on C ?
(1-1)^2＋(4-2)^2=4, so A on C
L: y-4=m(x-1)
mx-y-m+4=0
O=(1,2), r=2
d(O,L)=|m-2-m+4|/sqrt(m^2+1)=2
2/sqrt(m^2+1)=2
2=2 * sqrt(m^2+1)
m=0
L: y-4=0
方法二. 直線的向量與半徑的向量內積為零 ( L.r=0 )
vector L=(x-1,y-4)
vector r=A-O=(0,2)
L.r=0
2(y-4)=0
y-4=0
2.過B(2,4)求圓的切線(check B doesn't on C)
方法一.點到直線距離=r ( d(O,L)=r )
L: y-4=m(x-2)
mx-y-2m+4=0
O=(1,2), r=2
d(O,L)=|m-2-2m+4|/sqrt(m^2+1)=2
|-m+2|=2*sqrt(m^2+1)
m^2-4m+4=4(m^2+1)
3m^2+4m=0
m=0,-4/3
y-4=0
or
y-4=-4/3(x-2)
4x+3y-8-12=0
4x+3y-20=0
方法二. 直線的單位法向量與半徑的向量內積=半徑 ( n.r=半徑 )
L: y-4=m(x-2)
mx-y-2m+4=0
vector n=(m,-1)/sqrt(m^2+1)
vector r=(x-1,y-2)
n.r=2
|m(x-1)-(y-2)|/sqrt(m^2+1)=2
|mx-y-m+2|=2*sqrt(m^2+1)
|2m-4-m-2|=2*sqrt(m^2+1)
|m-2|^2=4(m^2+1)
m^2-4m+4=4m^2+4
3m^2+4m=0
m=0,-4/3
y-4=0
or
y-4=-4/3(x-2)
4x+3y-8-12=0
4x+3y-20=0
