Mark the tick against the correct answer in the following:

According to the question,

$\mathrm{O}$ is the origin

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

Formula

For a relation $R$ in set $A$

Reflexive

The relation is reflexive if $(a, a) \in R$ for every $a \in A$

Symmetric

The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$

Transitive

Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$

Equivalence

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

Check for reflexive

Consider, $(P, P) \in R \Leftrightarrow O P=O P$

which is always true.

Therefore , R is reflexive ……. (1)

Check for symmetric

$(P, Q) \in R \Leftrightarrow O P=O Q$

$(Q, P) \in R \Leftrightarrow O Q=O P$

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

Therefore , R is symmetric ……. (2)

Check for transitive

$(P, Q) \in R \Leftrightarrow O P=O Q$

$(Q, R) \in R \Leftrightarrow O Q=O R$

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

Also,

$(P, R) \in R \Leftrightarrow O P=O R$

$\therefore$ It will always be true

Therefore , R is transitive ……. (3)

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

Correct option will be (D)