A tricky but silly doubt regarding the solutions of $x^2/(y-1)^2=1$ Motivation : 
I have been confused with some degree 2 equation. I suddenly came across a simple equation and couldn't get the quintessence behind that. 
I have an equation $$\dfrac{x^2}{(y-1)^2}=1 \tag{1}$$ and I was looking for its solutions. It was asked by some kid ( of $9$nth standard ) to me. 
I did some manipulation and got $$x^2=(y-1)^2 \tag{2} $$ finally. And one can see that $(0,1)$ satisfies the Equation $[2]$ well. But I was happy, and within small time, 
I realized that the same solution set can't satisfy the equation $[1]$ . If you substitute $(0,1)$ in $[1]$ you get $\dfrac{0}{0}=1$ which is wrong.
The  answer that convinced me finally : 
We can see the same equation as this $x^2. \dfrac{1}{(y-1)^2}=1$ . We know that the set  of integers form a ring. So the product of two numbers is one if one number is the inverse of other number. '$1$' present on the R.H.S is the identity element. So the product of the entity with its inverse always gives us the identity. 
So when $x$ is $0$, the $0$ doesn't have an inverse in the integers. So the case is to be emitted. 
Still persisting questions :
But the thing that makes me surprise is that the Wolfram Alpha gives me this solution . 
                   
In the picture you can clearly see that they both intersect at $(0,1)$ . But what is that confusion ? 
We omitted that solution, but in fact $(0,1)$ is the intersection of the two lines. 
Questions that are to be answered by learned people : 


*

*What is the value of term $\dfrac{0}{0}$ ? Isn't it $1$ ? 

*Why  the solution pair $(0,1)$  satisfies $x^2=(y-1)^2$ but not $\dfrac{x^2}{(y-1)^2}=1$ ? We know that both of them are manifestations of each other in a simple manner. 

*If we need to omit that solution, why do the lines intersect at $(0,1)$ ? 
Thank you everyone for giving your time. 

You cannot multiply both side of an equation with $0$. When you multiply your equation by $(y-1)^2$, you are assuring that you are not taking $y=1$ as a valid solution.
