Thanks!
For an equation of nth degree given by
$\displaystyle ax^n +b* x^{n-1} + c*x^{n-2}..... = 0 $
The sum of all its roots
$\displaystyle x_1 + x_2 + x_3 ...+x_n = -b/a$
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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
The derivative of $\displaystyle y= x^2$ is y'= 2x. In particular, the derivative at $\displaystyle (x_0, x_0^2)$ is $\displaystyle 2x_0$ and so the equation of the normal line is $\displaystyle y= -\frac{1}{2x_0}(x- x_0)+ x_0^2$. The three normal lines, then, are
$\displaystyle y= -\frac{1}{2x_0}(x- x_0)+ x_0^2$
$\displaystyle y= -\frac{1}{2x_1}(x- x_1)+ x_1^2$
and
$\displaystyle y= -\frac{1}{2x_2}(x- x_2)+ x_2^2$
Saying that the all intersect at a common point means there exist a specific x, say $\displaystyle \overline{x}$ such that
$\displaystyle -\frac{1}{2x_0}(\overline{x}- x_0)+ x_0^2= -\frac{1}{2x_1}(\overline{x}- x_1)+ x_1^2$ and
$\displaystyle -\frac{1}{2x_0}(\overline{x}- x_0)+ x_0^2= -\frac{1}{2x_2}(\overline{x}- x_2)+ x_2^2$
Find the conditions on $\displaystyle x_0$, $\displaystyle x_1$, and $\displaystyle x_2$ so that those two equations have a common solution.
I will make myself clearer...its another way of doing your problem.
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For a parabola
$\displaystyle x^2 = 4ay$
the parametric equation of its normal at any general point (2at,at^2) is
$\displaystyle (y-at^2) = \frac{-1}{4at} (x-2at)$
$\displaystyle (-4ayt+4a^2t^3) = (x-2at)$
$\displaystyle -4ayt+4a^2t^3 -x+2at = 0$
$\displaystyle 4a^2t^3~+~ 0*t^2 ~+(2-4y)at~-x = 0$
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 .
ie;
$\displaystyle t_1 + t_2 +t_3 = 0$ (as given by first post of mine)
This implies
$\displaystyle 2at_1 + 2at_2 + 2at_3 = 0 $
But $\displaystyle 2at_1 = x_1$ and so on
Thus
$\displaystyle x_1 + x_2 + x_3 = 0 $
In your question it is to be proved for $\displaystyle a = 1/4$ but here we have proved it for all a.
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