I'm trying to prove that the focus of the equation to the parabola y^2=x is at (.25,0); for y^2=x, the "p" value is .25. My hypothesis is that all normals of the lines tangent to each ordered pair for the equation y^2=x intersect at the focus.
I used 3 points on y^2=x to demonstrate this; Points A,B,C, [(1,1),(4,2),(9,3)] respectively.
I used the first derivative function to find the slops at each point: y'(x) = (.5)x^(-.5)
@ A: Slope = .5
@ B: Slope = .25
@ C: Slope = 15/90
Tangent Lines through A,B,C are (respectively):
y = .5x + .5
y = .25x +1
y = (15/90)x + 1.5
The corresponding equations for the normals are (respectively):
y = -2x+3
y = -4x+18
y = -6x+27
The parabola opens to the right and has the focus located at (.25,0). If my hypothesis and calculations are done correctly, all 3 normal equations should intersect at the focus. Thus (.25,0) should be the ordered pair for all three equations of the normals. But it isn't.
[Solved: No connection between parabolic reflection and normals]