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The lines (x-2)/(1)=(y-3)/(2)=(z-4)/(0) ...

The lines `(x-2)/(1)=(y-3)/(2)=(z-4)/(0)` is

A

parallel to x-axis

B

parallel to y-axis

C

parallel to z-axis

D

lies in a plane parallel to xy-plane

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given line equation in 3D space, which is represented as: \[ \frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z - 4}{0} \] ### Step 1: Identify the Components of the Line Equation The given equation can be rewritten in a more standard form. We can denote: - \( x_1 = 2 \) - \( y_1 = 3 \) - \( z_1 = 4 \) The direction ratios of the line can be extracted from the denominators of the fractions: - Direction ratios: \( a = 1 \), \( b = 2 \), \( c = 0 \) ### Step 2: Analyze the Direction Ratios The direction ratios indicate the direction in which the line extends in 3D space. Here, since \( c = 0 \), it means that there is no change in the z-coordinate as we move along the line. This implies that the line does not rise or fall in the z-direction. ### Step 3: Determine the Orientation of the Line Since the z-component of the direction ratios is zero, the line is: - **Perpendicular to the z-axis**: This means it does not move in the z-direction. - **Parallel to the xy-plane**: The line lies flat in the xy-plane. ### Step 4: Conclusion Based on the analysis, we can conclude that the line described by the equation: \[ \frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z - 4}{0} \] is a line that lies in a plane parallel to the xy-plane. ### Final Answer The line lies in a plane parallel to the xy-plane. ---
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