Show that the relation R in the set A of points 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

Solution:

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

Clearly, ( P, P) ∈ R

∴ R is reflexive.

(P, Q) ∈ R

Clearly, R is symmetric.

(P, Q), (Q, S) ∈ R

⇒ The distance of P and Q from the origin is the same and also, the distance of Q and S

from the origin is the same.

⇒ The distance of P and S from the origin is the same.

(P, S) ∈ R

∴ R is transitive.

R is an equivalence relation.

The set of points related to P ≥ (0, 0) will be those points whose distance from the origin is the same as the distance of P from the origin.

A set of points forms a circle with the centre as the origin and this circle passes through P

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 11

Show that the relation R in the set A of points 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 the centre

Summary:

Hence we have shown that the relation R in the set A of points 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