Question

The plane $ x - 2y + z = 0 $ is parallel to the line

• A. \( \frac{x-3}{4} = \frac{y-4}{5} = \frac{z-3}{6} \)
• B. \( \frac{x-2}{4} = \frac{y-7}{5} = \frac{z-3}{7} \)
• C. \( \frac{x-2}{3} = \frac{y-3}{3} = \frac{z-4}{4} \)
• D. \( \frac{x-4}{3} = \frac{y-5}{4} = \frac{z-6}{3} \)

Hint:

To identify if two lines are parallel, check if their direction ratios are proportional. The general form of the equation for a line is \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \), where \( (a, b, c) \) is the direction ratio.

Answer 1

The Correct Option is A

To find the equation of a plane parallel to the given plane, we need to check the direction ratios. 
The direction ratios of the given plane \( x - 2y + z = 0 \) are the coefficients of \( x, y, \) and \( z \), which are \( (1, -2, 1) \). 
This gives us the direction of the line. Now, we are looking for the equation of the line that is parallel to the plane, so we need to find an equation whose direction ratios match the given plane. 
Let us check each option: 
- Option (A) has the direction ratios \( (4, 5, 6) \), which are proportional to \( (1, -2, 1) \). So, this option satisfies the condition. 
- Option (B) has the direction ratios \( (4, 5, 7) \), which are not proportional to \( (1, -2, 1) \), hence it does not satisfy the condition. 
- Option (C) has the direction ratios \( (3, 3, 4) \), which are not proportional to \( (1, -2, 1) \), so this option does not satisfy the condition. 
- Option (D) has the direction ratios \( (3, 4, 3) \), which again are not proportional to \( (1, -2, 1) \), so it does not satisfy the condition. 
Therefore, the correct answer is option (A).