parabola passes through $4$ points

Question

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

Answer 1

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?)

Answer 2

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.