For an equation of nth degree given by
The sum of all its roots
I hope you know the equation of normal passing through a point on parabola.
In that third degree equation
x^2 term will be missing... this tells you that
the value of its coefficient is 0
Hence the sum of roots= (X1+X2+X3) = 0 /a = 0
Saying that the all intersect at a common point means there exist a specific x, say such that
Find the conditions on , , and so that those two equations have a common solution.
I will make myself clearer...its another way of doing your problem.
For a parabola
the parametric equation of its normal at any general point (2at,at^2) is
This is a third degree equation....
Three different values of t satisfy the above equation ....
This means normals at these three different t intersect at (x,y).
(This can further be interpreted as your question )
Now the sum of three roots is zero .
(as given by first post of mine)
But and so on
In your question it is to be proved for but here we have proved it for all a.
I love math done beautifully...Thank you very much!