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Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1 , L2 ) : L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.



Class 12, NCERT Chapter 1,  Exercise 1.1, Q14


Line L is parallel to itself. This means L is parallel to L.
So, (L1, L2)∈R
So, R is reflexive. 
Let (L1, L2) ∈R
Then L1 is parallel to L2. So, L2 is parallel to L1.
This means, (L2, L1) ∈ R
So R is symmetric.
Let (L1, L2) ∈ R and (L2, L3) ∈ R
So, L1 is parallel to L2 and L2 is parallel to L3.
This means L1 is parallel to L3
So R is transitive.
So the R is equivalence Relation.

Find the set of all lines related to the line y = 2x + 4.
R = {(Li, L2) : Li is parallel to L2}
Set of all lines related to y = 2x + 4,
is set of all lines that are parallel to y = 2x + 4.

Let equation of line parallel to y = 2x + 4 be
 y = mx + c , where m is the slope of line

Since y = 2x + 4 & y = mx + c are parallel,
Slope of (y = 2x + 4) = Slope of (y = 2x + 4)
2 = m i.e. m = 2                                                                   (Slope of y = 2x + 4 is 2)

Hence, the required line is
 y = mx + c
i.e.y=2x+c, where c ∈ R. 
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