Question

The equation of X-axis is.
\[(a)\,\dfrac{X}{1}=\dfrac{Y}{0}=\dfrac{Z}{0}\]
\[(b)\,\dfrac{X}{0}=\dfrac{Y}{1}=\dfrac{Z}{1}\]
\[(c)\,\dfrac{X}{1}=\dfrac{Y}{1}=\dfrac{Z}{1}\]
(d) None of these

Answer 1

Hint: In this question, we have to write the equation of x-axis in 3d plane. The basic equation of any line is represented by the equation: \[\,\dfrac{x-{{x}_{1}}}{a}=\dfrac{y-{{y}_{1}}}{b}=\dfrac{z-{{z}_{1}}}{c}=r\].

Complete Complete step by step answer:
We can represent any point(P) on the line as follows: \[P\left( {{x}_{1}}+ar,{{y}_{1}}+br,{{z}_{1}}+cr \right)\], where ‘r’ can have any value. The vector form of the line can be represented as: \[\vec{z}=\vec{p}+r\vec{q}\].

Now, we can start the solution as we have enough information. As we have to write the equation of x-axis in 3d, we must know the points through which it passes. We can determine that point easily. As we know, it is an x-axis, the y-component and z-component would always be zero. We can also say that the x-axis passes through the origin (0,0,0). Also, let us consider the values of x as 1, then another point through which the x-axis passes can be (1,0,0). So, now we have two points: \[{{x}_{1}}\left( 1,0,0 \right)\] and \[{{x}_{2}}\left( 0,0,0 \right)\]. Now, we can substitute the points in the equation of a line. So, the equation of the x-axis becomes:
\[\,\dfrac{x-0}{1-0}=\dfrac{y-0}{0-0}=\dfrac{z-0}{0-0}\]
After simplification, we get:
\[\begin{align}
  & \,\dfrac{x}{1}=\dfrac{y}{0}=\dfrac{z}{0} \\
 & \Rightarrow \dfrac{X}{1}=\dfrac{Y}{0}=\dfrac{Z}{0} \\
\end{align}\]
From the above options, option (a) matches our answer. Hence, option (a) is correct.

Note: The alternate method for solving these types of questions is checking the options one by one and eliminating the wrong ones. This method is faster but we should know the points that satisfy the line. Since we have to find the equation of x-axis, we can easily choose option (a) as the correct answer because y and z coordinates would be zero.