Finding normals to a curve through a given point

Question

I want to find x on curve $y(x)$, where curve $y(x)$ is perpendicular to point $(x_0,y_0)$. for this I minimize the distance between points $(x,y(x))$ and $(x_0,y_0)$, here's an example for point $x_0=4,y_0=3$.

y[x_] := 2 + 0.36 x
NMinimize[(4 - x)^2 + (3 - y[x])^2, x]

But somehow this does not give the right answer though Mathematica seems to be finding the right minimum.

{0.171388, {x -> 3.85977}}

Update:

In a more practical case, with Ei some data to be fitted, and sol2 the fitted coefficients of U[...] (refer to the end of the question for detail), this is what I tried:

Do[
 int = FindMinimum[{(Ei[1][[1, 2]] - U[x, 1] /. sol2[[2]])^2 + (i - x)^2}, x];
 AppendTo[perpendicular, int], 
 {i, 1, 30}]

cc = 
  ListPlot[{
    Evaluate @ Table[{i, Ei[1][[i, 2]]}, {i, 15, 23}], 
    Evaluate @
      Table[{x, U[x, 1] /. sol2[[2]]} /. perpendicular[[i, 2]], {i, 15, 23}]}, 
    Filling -> {1 -> {2}}, 
    PlotRange -> {0, 5}, 
    AspectRatio -> Automatic];

qq = 
  Plot[U[x, 1] /. sol2[[2]], {x, 15, 23}, 
    PlotRange -> {0, 5}, 
    AspectRatio -> Automatic];

This is the fitting function:

U[r_, o_] := 
  Sum[-ehh (1 - (1 - Exp[-Ahh (raa[i, j, o] - rshh)])^2), {i, 2, 
      5}, {j, 7, 10}] + 
    Sum[-ech (1 - (1 - Exp[-Ach (raa[i, j, o] - rsch)])^2), {i, 1, 
      1}, {j, 7, 10}] + 
    Sum[-ech (1 - (1 - Exp[-Ach (raa[i, j, o] - rsch)])^2), {i, 2, 
      5}, {j, 6, 6}] - 
    ecc (1 - (1 - Exp[-Acc (raa[1, 6, o] - rscc)])^2) /. x -> r;

And this the the fitted coefficients:

sol2 = {ehh -> -4.07603, Ahh -> 3.75309, rshh -> 1.44794, ech -> 0.223024, 
        Ach -> 1.51935, rsch -> 2.84276, ecc -> -4.52077, Acc -> 2.99396, 
        rscc -> 2.60361}

Answer 1

If you make a line from blue and red points on the curve you will see they all are kinda pointing at one point.

There I found a mistake at your code, I think Ei[1][[1 , 2]] this is a static number, I guess it should be a various number changing by iterator i so you're finding minimum b=distance between 2 different points each time.

Do[
 int = FindMinimum[{(Ei[1][[i, 2]] - U[x, 1] /. sol2[[2]])^2 + (i - x)^2}, x];
 AppendTo[perpendicular, int], 
 {i, 1, 30}]

Answer 2

p = {x, y[x]} /. Last@NMinimize[(4 - x)^2 + (3 - y[x])^2, x]
(* {3.85977, 3.38952} *)

Dot[({4, 3} - p ), ({0, y[0]} - {5, y[5]})]

(* -3.9641*10^-8 *)

Numerically zero. As others said it doesnt look right on your plot because of the aspect ratio of the plot, try this:

Show[{ Plot[ y[x] , {x, 0, 5} ] ,  Graphics[Line[ {{4, 3}, p}]]}, 
       AspectRatio -> Automatic]