Find the equivalence classes of $[(0,0)],[(1,0)],[\left(\sqrt2,0\right)]$ and $[(0,-2)]$

Question

On the set $\mathbb{R}^2$, define $(x,y) R (a,b)$ if and only if $x^2-y =a^2-b$. Find the equivalence classes of $[(0,0)],[(1,0)],[\left(\sqrt2,0\right)]$ and $[(0,-2)] $and give a geometric description of them.

---

$[(0,0)]=\{(x,y)\in\mathbb{R^2}:(x,y)R(0,0)\}=\{(x,y)\in\mathbb{R^2}:x^2-y=0\}$

$[(1,0)]=\{(x,y)\in\mathbb{R^2}:(x,y)R(1,0)\}=\{(x,y)\in\mathbb{R^2}:x^2-y=1\}$

$[(0,1)]=\{(x,y)\in\mathbb{R^2}:(x,y)R(0,1)\}=\{(x,y)\in\mathbb{R^2}:x^2-y=-1\}$

$[(\sqrt2,0)]=\{(x,y)\in\mathbb{R^2}:(x,y)R(\sqrt2,0)\}=\{(x,y)\in\mathbb{R^2}:x^2-y=2\}$

$[(0,2)]=\{(x,y)\in\mathbb{R^2}:(x,y)R(0,2)\}=\{(x,y)\in\mathbb{R^2}:x^2-y=-2\}$

Geometric description: parabolas of the form $y=x^2-a^2+b$

Is my work correct?

Answer 1

Yes, your working is fine.

The equivalence class of $[(a,b)]$ is the parabolas of the form of $y=x^2-a^2+b$ where $-a^2+b$ is the intercept term.