Question

Which of the following cannot be the direction ratios of the straight line \(\frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1}\)?

• A. \(2, -3, -1\)
• B. \(-2, 3, 1\)
• C. \(2, 3, -1\)
• D. \(6, -9, -3\)

Hint:

When finding the direction ratios of a line from its parametric form, look for the coefficients of \(x\), \(y\), and \(z\). These coefficients directly give you the direction ratios. Additionally, any scalar multiple of these direction ratios represents a line with the same direction, so negatives and multiples are valid answers. Always verify the sign and magnitude of direction ratios before concluding.

Answer 1

The Correct Option is C

The direction ratios of a straight line are the coefficients of x, y, and z in the parametric form of the line.

The given equation of the line is:

This can be rewritten in parametric form as:

x = 3 + 2t, y = 2 - 3t, z = -4 - t.

Thus, the direction ratios are 2, -3, -1.

Now, check each option:

Option (1) 2, -3, -1 is correct since it matches the direction ratios.

Option (2) -2, 3, 1 is a negative multiple of the correct direction ratios, so it is valid.

Option (3) 2, 3, -1 does not match the direction ratios, as the second direction ratio should be negative.

Option (4) 6, -9, -3 is a positive multiple of the direction ratios, so this is valid.

Thus, the correct answer is:

2, -3, -1.

Answer 2

The direction ratios of a straight line are the coefficients of \( x \), \( y \), and \( z \) in the parametric form of the line.

The given equation of the line is:

This can be rewritten in parametric form as:

\[ x = 3 + 2t, \quad y = 2 - 3t, \quad z = -4 - t. \]

Thus, the direction ratios are: \( 2, -3, -1 \).

Now, check each option:

Option (1): \( 2, -3, -1 \) is correct since it matches the direction ratios.

Option (2): \( -2, 3, 1 \) is a negative multiple of the correct direction ratios, so it is valid.

Option (3): \( 2, 3, -1 \) does not match the direction ratios, as the second direction ratio should be negative.

Option (4): \( 6, -9, -3 \) is a positive multiple of the direction ratios, so this is valid.

Thus, the correct answer is:

2, -3, -1.