Question

The Cartesian equations of the line passing through \(A(0,1,1)\) and parallel to the X-axis are ....

• A. \(y = 1,z = 1\)
• B. \(x = 0\)
• C. \(x+y+z = 2\)
• D. \(y = z\)

Hint:

A line parallel to the x-axis keeps y and z fixed.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A line parallel to the X-axis has direction ratios \((1,0,0)\). Along such a line only \(x\) changes.

Step 2: Write the equations
The line passes through \(A(0,1,1)\), so
\[ \frac{x-0}{1}=\frac{y-1}{0}=\frac{z-1}{0} \]
This means \(y-1=0\) and \(z-1=0\), that is \(y=1,\ z=1\).

Step 3: Check the options
(B) \(x=0\) is a plane, not a line. (C) \(x+y+z=2\) is a plane. (D) \(y=z\) is also a plane. Only (A) gives two plane equations that together define the line.

Final Answer:
A line parallel to the X-axis through (0, 1, 1) is y = 1 and z = 1, option (A). \[ \boxed{y=1,\ z=1} \]