Intersection of parabola and vertical line at a particular point

Question

I want to create a parabola that intersects a vertical line at a particular point. Can someone help me with this?

my MWE is

\usepackage{tikz}
\usetikzlibrary{intersections,shapes,arrows,calc}
\usepackage{esvect}
\usepackage{relsize}

\begin{tikzpicture}
% New graph - bottow left 
\draw[<->,rounded corners, thick]  (0,2.5) node(yline)[left] {$r$} -- (0,0) -- (3,0)  node(yline)[below] { $Y$}; 
\draw (1.5,0) -- (1.5,2.5);
\node[label=right:$\overline{S}(Y_{A})$] at (1.5,2.5) {};
\node[ circle,fill=black,minimum size=2pt,inner sep=0pt, outer sep=-1pt, label=left:$r^*_{A}$] at (0,1.5) {};
\draw[dashed]  (0,1.5) -- (1.5,1.5) ;
\draw (0.8,2.5) parabola[bend at end] (1.5,1.5) ;
\draw (1.5,1.5) parabola[bend at end] (2.5,1.2) ;
\end{tikzpicture}

Which produces the following agly picture:

I am very rigid about the intersection point. In my case the parabola has to pass through the (1.5,1.5) point

Any help to make the graph look better ?

Answer 1

See, if you liked ...

\documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{intersections}

\begin{document}
\begin{tikzpicture}
% axes
\draw[thick,->]  (-0.1,0) coordinate (O)
                          -- ++ (4.1,0) node [below left] {$X$};
\draw[thick,->]  (0,-0.1) -- ++ (0,4.1) node [below left] {$r$};
\draw[ultra thin, gray] (0,0) grid + (4,4); %  only to show that intersection is at (1.5,1.5)
% curve
\begin{scope}[font=\footnotesize]
\draw[red, very thick,name path=A] 
    plot[domain=0.32:4, samples=64] (\x,0.8333+1/\x);
\draw[name path=B]   
                (1.5,-0.1) node[below] {1.5}
                           -- ++ (0,4.1) node[below right] {$\overline{S}(Y_{A})$};
\draw[dashed, name intersections={of =A and B, by={r}}] 
    (r) -- (r -| O) node[left]  {$r^*_{A}$};
\end{scope}
\end{tikzpicture}
\end{document}

Addendum: In the first version of answer I assume, that the main problem is redesign LaTeX part of image, e.g. defining of function is not problem. It seems, that I was wrong, so now I correcting this my faulty performance :) but on the way of minimal effort. You can find adequate function yourself, and of course change tick labels to what you like to have. Added grid is only indicative and had to be removed in real use of MWE (as I understand your wish).

Note: shoved curve is not parabola, I only mimic what you shown in your question with inverse function i.e. hyperbole.