parabola passes through $4$ points Finding maximum number of parabola which passes through the point $A(1,2)\;,\; B(2,1)\;\;,(3,4)\;\;,(4,3)$


what i try

from these $4$ point one can imagine that parabola symmetrical about $y=x$ line

so axis of parabola along the line $y=x$ and directrix along the line $x+y+c=0$, where $c$ is any constant

did not understand how do i solve further

help me to solve it please
You rightly noticed that the four points are symmetrical around $y=x$.
So change the variables accordingly
$$
\left\{ \matrix{
  \xi  = y - x \hfill \cr 
  \eta  = y + x \hfill \cr}  \right.\quad  \Leftrightarrow \quad \left\{ \matrix{
  x = {{\eta  - \xi } \over 2} \hfill \cr 
  y = {{\eta  + \xi } \over 2} \hfill \cr}  \right.
$$

The four points become
$$
\left( {1,3} \right),\left( { - 1,3} \right),\left( {1,7} \right),\left( { - 1,7} \right)
$$

The parabola's axis is the $\eta$ axis, so its formula is
$$
\eta  - a = k\xi ^{\,2} 
$$

In any case it is clear that, since the points form a rectangle, there cannot be a single parabola passing through all of them.
Hint:

A parabola is of the form $y = ax^2 + bx + c$

We are told that it passes through the points $(1,2),(2,1)$ and $(3,4)$.  (I suggest ignoring $(4,3)$ for now and revisit it later)

That means that plugging in those $x$ and $y$ values, the fact that it passes through $(\color{blue}{1},\color{red}{2})$ implies that $\color{red}{2} = a\cdot \color{blue}{1}^2 + b\cdot \color{blue}{1} + c$.  Similarly we can find the equations that are implied by the parabola passing through the other points.

We have then this system of equations:

$$\begin{cases}2 = a + b + c\\1 = 4a + 2b + c\\ 4 = 9a + 3b + c\\\end{cases}$$

Now... this is a system of three linear equations and three unknowns which can by solved by standard means.

(Now, consider including $(4,3)$ as well.  Does this affect the answer?)