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 equation of the line is given by the symmetric equation: 

\(\frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1} = k\), where \(k\) is a parameter.

This represents a line in three-dimensional space, with the direction ratios provided by the denominators of each fraction (i.e., coefficients of \((x-3)\), \((2-y)\), and \((z+4)\)) as they relate to the parameter \(k\). These direction ratios are \(2\), \(-3\), and \(-1\).

The direction ratios of a line in three dimensions can be arbitrarily scaled. Therefore, any set of direction ratios is valid if it is a scalar multiple of \((2, -3, -1)\).

Let's examine the given options:

1. \(2, -3, -1\): This is the original set of direction ratios.
2. \(-2, 3, 1\): This is a scalar multiple of the original by \(-1\).
3. \(2, 3, -1\): This cannot be a scalar multiple of the original because changing the sign of only one component breaks the proportional relationship that defines direction ratios.
4. \(6, -9, -3\): This is a scalar multiple of the original by \(3\).

Thus, the option \(2, -3, -1\) cannot be the direction ratios of the line as it doesn't maintain the required proportional transformations that can exist between direction ratios. This inconsistent alteration indicates it does not align with the line's direction as derived from the given symmetric equation.

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.