I had read in my notes that the equation of parabola can be given by
(Equation of axis)$^2$ = (Length of Latus rectum)*(Equation of tangent at vertex)
(I don't know the systematic proof. Is there something I am missing in the equation?)
Now take look at this very basic equation of a parabola
$ y^2=4ax $
Here the equation of axis of parabola is $(y=0)$ and that of tangent at vertex is $(x=0)$
I can also write the equation of axis as $(ny=0)$ and tangent at vertex as $(mx=0)$
(where m and n are constants)
And hence using the first equation I can write the equation of parabola as
$(ny)^2 = 4a(mx)$
which gives me a completely different parabola.
I don't know where I have gone wrong. Please guide me.
Best Answer
Here, it is not equation of axis, it is perpendicular distance of any random point on parabola say (h,k) from axis and also perpendicular distance from vertex and not equation of vertex So for general equation Perpendicular distance from axis of point (h,k) is k and from vertex, it's h So equation will be:
k²=4ah
Replacing h with x and k with y, we get equation of parabola y²=4ax