# 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