# Geo5.1.42

From Example Problems

Prove that the locus of the point of intersection of two perpendicular normals to the parabola is the parabola

Let be the ppoint of intersection of the normals at

Equation of the normal at t is

If it passes through (x1,y1)

If t1,t2,t3 are the roots of the above equation, .

Let this be equation 2

Slope of the normal at t1=-t1.

Slope of the normal at t2=-t2.

Given PA is perpendicular to PB .equation 3

From 2 and 3

Since the normal at t3 also passes thro'

. equation 4.

Hence the locus of P is