Equation of parabola passes through $4$ distinct points 
Equation of axis of parabola which


passes through the point $(0,1)\ , \ (0,2)$


And $(2,0)\ ,\ (2,2)$ is

Let general equation of conic is
$ax^2+2hxy+by^2+2gx+2fy+c=0\cdots (1)$
And it represent parabola if $h^2=ab$
Parabola passes through $(0,1)$
Then put into $(1)$
$b+2f+c=0\cdots(2)$
Also parabola passes through $(0,2)$
Then $4b+4f+c=0\cdots (3)$
Also parabola passes through $(2,0)$
Then $4a+4g+c=0\cdots (4)$
Also parabola passes through $(2,2)$
Then $4a+8h+4b+4g+4f+c\cdots (5)$
$(4a+4g+c)+(4b+4f+c)+8h-c=0$
From $(2)$ and $(3)$, we get $\displaystyle h=\frac {c}{8}$
From $(2)$ and $(3)$
$\displaystyle b+2f=4b+4f\Longrightarrow 2f=-3b $
Put into $(2)$ and $(3)$
$\displaystyle b=\frac{c}{2}$ and $\displaystyle f=-\frac{3c}{4}$
And $\displaystyle h^2=ab\Longrightarrow \frac{c^2}{64}=\frac{ac}{2}\Longrightarrow a=\frac{c}{32}$
Put all into $(4)$
$\displaystyle \frac{4c}{32}+4g+c=0\Longrightarrow g=-\frac{9c}{32}$
Put all values into $(1)$
$\displaystyle \frac{c}{32}x^2+\frac{c}{4}xy+\frac{c}{2}y^2-\frac{9}{16}x-\frac{3c}{2}y+c=0$
$x^2+8xy+16y^2-18x-48y+32=0$
Buti did not know how I find axis of parabola
Please have a look
 A: What you've done is correct.
One way to find the axis is to write the equation as $(x+4y)^2-18x-48y+32=0,$ then you know the axis is parallel to $x+4y=0.$ The tangent at vertex is orthogonal and parallel to $4x-y.$ Now you can find where $4x-y+c$ intersects doubly to find the vertex. $$x^2+8x(4x+c)+16(4x+c)^2-18x-48(4x+c)+32=0$$ or $$289x^2+(136c-210)x+16c^2-48c+32=0$$ and this has discriminant $$(136c-210)^2-4\cdot 289\cdot (16c^2-4c+32)=(-4)(408c - 1777)$$ so the discriminant is zero for $c=\frac{1777}{408}.$ The vertex is the intersection of the tangent at vertex and the parabola. So solve $$x^2+8x(4x+\frac{1777}{408})+16(4x+\frac{1777}{408})^2-18x-48(4x+\frac{1777}{408})+32=0$$ or $$(1734x+1147)^2\frac1{10404}$$ and put back into the tangent at vertex to find $y.$
The vertex then is $(-\frac{1147}{1734},\frac{11857}{6936}).$
The axis goes through the vertex  so the axis is $${ x+4y=\frac{105}{17}}.$$ As a bonus the tangent at vertex form is then $$(x+4y-105/17)^2=(96/(4\cdot 17))(4x-y+1777/408).$$ For fun I've also found the focus/directrix form $$17\cdot ((x+\frac{59}{102})^2+(y-\frac{689}{408})^2-(4x-y+\frac{113}{24})^2/17)=0$$

A: I'm going to show a solution which uses the following claim, and then add a proof of the claim and an important fact which might interest you.
Claim : The equation of a parabola can be written as
$$\bigg(f(x,y)\bigg)^2+g(x,y)=0$$
where
$f(x,y)=0$ is the equation of the axis of symmetry, and $g(x,y)=0$ is the equation of the tangent at vertex. (Note that the axis of symmetry is perpendicular to the tangent at vertex.)

Solution :
You correctly got
$$x^2+8xy+16y^2-18x-48y+32=0$$
which can be written as
$$(x+4y+c)^2+(-2c-18)x+(-8c-48)y-c^2+32=0$$
Since we want to find $c$ such that the line $x+4y+c=0$ is perpendicular to the line $(-2c-18)x+(-8c-48)y-c^2+32=0$, solving
$$1\times (-2c-18)+4\times (-8c-48)=0$$
gives $c=-\dfrac{105}{17}$, and so the equation of the axis of symmetry is $x+4y-\dfrac{105}{17}=0$.

In the following, I'll add a proof of the claim.
Claim : The equation of a parabola can be written as
$$\bigg(f(x,y)\bigg)^2+g(x,y)=0$$
where
$f(x,y)=0$ is the equation of the axis of symmetry, and $g(x,y)=0$ is the equation of the tangent at vertex.
Proof :
The equation
$$ax^2+2bxy+cy^2+2dx+2fy+g=0\tag2$$
represents a parabola iff $ac=b^2$ and
$$\begin{vmatrix}
a & b & d \\
b & c & f \\
d & f & g \\
\end{vmatrix}\not=0\tag3$$
(see here)
Multiplying the both sides of $(2)$ by $a$, and letting $$A=a,C=b,D=2ad,E=2af,F=ag$$
we see that, in general, the equation of a parabola is given by
$$(Ax+Cy)^2+Dx+Ey+F=0\tag4$$
(where $(A,C)\not=(0,0)$, and $(3)\iff CD-AE\not=0$) which can be written as
$$\bigg(f(x,y)\bigg)^2+g(x,y)=0$$
where
$$\begin{align}f(x,y)&=Ax+Cy+\frac {AD+CE}{2(A^2+C^2)}
\\\\g(x,y)&=\frac{CD-AE}{A^2+C^2}(Cx-Ay+G)
\\\\G&=\frac{4F(A^2+C^2)^2-(AD+CE)^2}{4(A^2+C^2)(CD-AE)}\end{align}$$
Note that $f(x,y)=0$ is perpendicular to $g(x,y)=0$.
Now, we can see the followings by some calculations :

*

*The line $g(x,y)=0$ is tangent to the parabola $(4)$ at $P\bigg(H,\dfrac{CH+G}{A}\bigg)$ where $H=\dfrac{-2CG(A^2+C^2)-A(AD+EC)}{2(A^2+C^2)^2}$.


*The line $f(x,y)=0$ intersects the parabola $(4)$ only at $P$.
From these, we can say the followings :

*

*$f(x,y)=0$ is the equation of the axis of symmetry


*$g(x,y)=0$ is the equation of the tangent at the vertex


*$P$ is the vertex of the parabola.$\quad\blacksquare$

Finally, I'll add an important fact which might interest you.
The equation of a parabola
$$(Ax+Cy)^2+Dx+Ey+F=0$$
can be written as
$$\bigg(\frac{f(x,y)}{\sqrt{A^2+C^2}}\bigg)^2=\frac{AE-CD}{(A^2+C^2)^{\frac 32}}\cdot\frac{Cx-Ay+G}{\sqrt{A^2+C^2}}$$
where

*

*$\dfrac{|f(x,y)|}{\sqrt{A^2+C^2}}$ represents the distance from $(x,y)$ to the axis of symmetry


*$\dfrac{|AE-CD|}{(A^2+C^2)^{\frac 32}}$ represents the length of its latus rectum (see here)


*$\dfrac{|Cx-Ay+G|}{\sqrt{A^2+C^2}}$ represents the distance from $(x,y)$ to the tangent at the vertex.
