Question

The equation of line parallel to y– axis is
(a) \[x=a\]
(b) \[y=b\]
(c) \[x+y=c\]
(d) \[x=y+1\]

Answer 1

Hint: We solve this problem by taking the general equation of a line.
The general representation of line is given as \[y-b=m\left( x-a \right)+c\] where, \[m\] is the slope of the line and \[a,b,c\] is some constants.
We know that two parallel lines have the same slope and we use the condition that the slope of the y-axis is infinity and the equation of the y-axis is given as
\[x=0\]
We are asked to find the equation of line parallel to the y-axis.

Complete step by step answer:
Let us assume that the line equation a general line equation.
We know that the general representation of line is given as \[y-b=m\left( x-a \right)+c\] where, \[m\] is the slope of the line and \[a,b,c\] is some constants
So, let us assume the required line equation as
\[\Rightarrow y-b=m\left( x-a \right)+c.....equation(i)\]
Let us take the equation of y– axis as
\[\Rightarrow x=0\]
Let us try to represent the equation of y– axis as general equation of line then we get
\[\Rightarrow y-0=\dfrac{1}{0}\left( x-0 \right)+\dfrac{1}{0}\]
Let us assume that the slope of y– axis as \[{{m}_{1}}\]
Here, we can see that the slope is given as
 \[{{m}_{1}}=\dfrac{1}{0}\]
We know that \[\dfrac{1}{0}\] is an undetermined form.
But for our convenience let us assume that it is the slope of y-axis (just for understanding only).
Now, we know that two parallel lines have the same slope.
By using the above condition to the required line and y– axis we get
\[\begin{align}
  & \Rightarrow m={{m}_{1}} \\
 & \Rightarrow m=\dfrac{1}{0} \\
\end{align}\]
Now, by substituting this slope in the equation of line that is equation (i) then we get
\[\begin{align}
  & \Rightarrow y-b=\dfrac{1}{0}\left( x-a \right)+c \\
 & \Rightarrow x=a \\
\end{align}\]
Therefore, we can conclude that \[x=a\] is the equation of line parallel to y– Axis.

Note:
Students may misunderstand the representation of the equation of the y-axis as the general equation of a line.
We have the representation of the equation of the y-axis as
\[\Rightarrow y-0=\dfrac{1}{0}\left( x-0 \right)+\dfrac{1}{0}\]