Question

Let L be the set of all lines in XY plane and R be the relation in L defined as $R=\left\{ \left( {{L}_{1}},{{L}_{2}} \right):{{L}_{1}}\,is\,parallel\,to\,{{L}_{3}} \right\}$. If R is an equivalence relation, then the set of all lines related to the line $y=2x+4$ is:
(a) $y=2x+c,\,c\in R$
(b) $y=4x$
(c) $y=4$
(d) $y=-2x$

Answer 1

Hint: In this question, we will write the required set in builder form and then check all its condition step by step and use parallel line condition to solve the question.

Complete step-by-step solution -
In a given question, we have set L of all lines in the xy plane.
Relation R is defined on set L, which is given as, $R=\left\{ \left( {{L}_{1}},{{L}_{2}} \right):{{L}_{1}}\,is\,parallel\,to\,{{L}_{3}} \right\}$. Let us name the required set as A.
Then A is given as, $A=\left\{ \left( a,{{L}_{3}} \right)\in R \right\}$ where a is an element of set L and ${{L}_{3}}$ is a line $y=2x+4$.
Now, for an element to belong to set A, it must first belong to L, that is, it must be a line.
Here, all the given options are equations of lines, hence, they all belong to L.
Now, for a ordered pair $\left( a,{{L}_{3}} \right)$ to belong to R, where a is element of L, a and ${{L}_{3}}$ must be parallel lines.
We know that, condition for two lines in the same plane to be parallel to each other is that they must have equal slope. Now, slope of a line in slope intercept form that $y=mx+c$ is m.
Using this, the slope of the given line ${{L}_{3}}$ $y=2x+4$ will be 2.
Now, any line with slope 2 will be parallel to ${{L}_{3}}$, and hence will be related to ${{L}_{3}}$ with relation R. Therefore, any line with slope 2 will belong to A.
Therefore, $A=\left\{ set\,of\,all\,lines\,with\,slope\,2 \right\}$. Now, slope intercept form of any line with slope 2, that is m=2, will be $y=mx+c$.
$\Rightarrow y=2x+c$, where c is a constant which belongs to real numbers.
Hence, A is a set of all lines of the form $y=2x+c,c\in R$.
Hence, the correct option is (a).

Note: You can also directly check the answer from options as two lines are parallel only when coefficients of x and y of both lines are proportional and constant terms can have any value.