According to the question ,

O is the origin

R = {(P, Q) : OP = OQ }

Formula

For a relation R in set A

Reflexive

The relation is reflexive if (a , a) ∈ R for every a ∈ A

Symmetric

The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R

Transitive

Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R

Equivalence

If the relation is reflexive , symmetric and transitive , it is an equivalence relation.

Check for reflexive

Consider , (P , P) ∈ R ⇔ OP = OP

which is always true .

Therefore , R is reflexive ……. (1)

Check for symmetric

(P , Q) ∈ R ⇔ OP = OQ

(Q , P) ∈ R ⇔ OQ = OP

Both the equation are the same and therefore will always be true.

Therefore , R is symmetric ……. (2)

Check for transitive

(P , Q) ∈ R ⇔ OP = OQ

(Q , R) ∈ R ⇔ OQ = OR

On adding these both equations, we get , OP = OR

Also,

(P , R) ∈ R ⇔ OP = OR

∴ It will always be true

Therefore , R is transitive ……. (3)

Now , according to the equations (1) , (2) , (3)

Correct option will be (D)