Normal Lines to a Parabola
Normal Lines to a Parabola
The normal line to the graph of parabola at the point , is , or equivalently +t-=0. The number of real roots of a reduced cubic +pt+q depends on the sign of . So the number of normal lines to through depends on the sign of . If , there is one normal line to through ; if , there are two normal lines; and if , there are three. (For the cusp , the only normal line is .)
y=a
2
x
(t,a)
2
t
t≠0
y-a=-(x-t)
2
t
1
2at
3
t
1-2ay
2
2
a
x
2
2
a
3
t
4+27
3
p
2
q
y=a
2
x
(x,y)
g(x,y)=4+27
3
1-2ay
2
2
a
2
x
2
2
a
g(x,y)>0
y=a
2
x
(x,y)
g(x,y)=0,(x,y)≠0,
1
2a
g(x,y)<0
0,
1
2a
x=0