Point of normalcy to a parabola from a point
Finding all the points of normalcy to a parabola from a point.
Hint : a normal line at a point K (k,k²) on the parabola goes through a point on the axis M(0,k+0.5)
If A = (m,n) the equation for the 3 points of normalcy (k1, k2 and k3) is :
k^3 - (n-0.5)k - 0.5m = 0
The discriminant is 4p^3 + 27q^2
if D < 0 we have 3 real points else only one point is visible.
For a point A (m,n), n > 0.5 - (-27m^2/16)^(1/3)