Q.11 Show that the relation R in the set A of points in a plane given by
, is an equivalence relation. Further, show that the set of
all points related to a point is the circle passing through P with origin as
centre.
The distance of point P from origin is always same as distance of same point P from origin i.e.
R is reflexive.
Let i.e. distance of the point P from the origin is same as the distance of the point Q from the origin.
this is same as : distance of the point Q from the origin is same as the distance of the point P from the origin i.e.
R is symmetric.
Let and
i.e. distance of the point P from the origin is same as the distance of the point Q from the origin, and aslo distance of the point Q from the origin is same as the distance of the point S from the origin.
We can say that distance of point P,Q,S from origin is same.Means distance of point P from origin is same as distance of point S from origin i.e.
R is transitive.
Hence, R is an equivalence relation.
The set of all points related to a point are points whose distance from origin is same as distance of point P from origin.
In other words we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from origin.
Hence, these set of points form circle with centre as origin and this circle passes through point P.
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The distance of point P from the origin is always the same as the distance of same point P from origin i.e.
Let
i.e. the distance of the point P from the origin is the same as the distance of the point Q from the origin.
this is the same as distance of the point Q from the origin is the same as the distance of the point P from the origin i.e.
Let
and 
i.e. distance of the point P from the origin is same as the distance of the point Q from the origin, and also the distance of the point Q from the origin is same as the distance of the point S from the origin.
We can say that distance of point P, Q, S from the origin is the same. Means distance of point P from the origin is the same as the distance of point S from origin i.e.
Hence, R is an equivalence relation.
The set of all points related to a point
are points whose distance from the origin is the same as the distance of point P from the origin.
In other words, we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from the origin.
Hence, these set of points form circle with the centre as the origin and this circle passes through point P.