Show that the relation R in the set A of point in a plane given by R = {(P, Q)}: distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Let A be the set of points in a plane. 

R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}

∴ R = {(P, Q): |OP| = | OQ|, where O is the origin}

Reflexive :  As | OP | = | OP |

∴ (P, P) ∈ R ∀ P ∈ A

⇒ R is reflexive.

Symmetric : Let (P, Q) ∈ R ⇒ |OP| = |OQ|

⇒|OQ| = |OP| ⇒ (Q, P) ∈ R

R is symmetric.

Transitive: Let (P, Q) ∈ R and (Q, S) ∈ R 

⇒ |OP| = |OQ| and |OQ| = |OS|

⇒ |OP| = |OS| ⇒ (P, S) ∈ R 

∴ R is transitive. 

Hence R is an equivalence relation.

Let B be the set of points in a plane related to P ≠ 0

∴ B = {Q ∈ A : (Q, P) ∈ R} = {Q ∈ A : |OQ| = |OP|}

= {Q ∈ A : Q lies on a circle passing through P with centre O}.