Show that relation R on the set A of points on coordinate plane given by R={(P, Q):distance OP=OQ, where O is the origin}is an equivalence relation. Further find the set of all points related to a point Pnot equal to (0,0) is ylthe circle passing through P with origin as center

Question

12 Show that the oulation \( R \) on the set \( A \) of points on co-ordinate plane givem by \( R=\{(P, Q) \) : idistance \( O P=O Q \), where \( O \) is the origin ) is an equivalence velation. Furtier find the set of als points velated Uto apoint \( P \neq(0,0) \) is the circlepeissing throrgh \( p \) with the rigin as centre.

Answer 1

A is 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} = {(PQ) |OP| = |OQ | where O is origin} Since | OP | = | OPI, (PP) ERVPE A.

... R is reflexive.

Also (P. Q) ER ⇒|OP| = |OQ|

⇒ | OQ | = |OP|

⇒ (Q.P) ER⇒ R is symmetric. Next let (PQ) E R and (Q, T) ER⇒ |OP|=|OQ | and

OQ | = |OT|

⇒ |OP| = |OT|

→> (PT) ER

.. R is transitive.

... R is an equivalence relation.

Set of points related to P = 0

= {Q E A: (Q,P) E R} = {QEA: |OQ| = |OP|} = {Q E A :Q lies on a circle through P with centre O}.