Question

If the direction cosines of a line are \( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \), then

• A. \( 2<c<3 \)
• B. \( c = \pm 3 \)
• C. \( c = \pm \sqrt{3} \)
• D. \( c = \pm \frac{1}{\sqrt{3}} \)

Hint:

For direction cosines, remember the identity \( l_x^2 + l_y^2 + l_z^2 = 1 \) and use it to solve for the unknown values.

Answer 1

The Correct Option is C

Step 1: Use the condition for direction cosines.
The direction cosines of a line \( l \) are given by \( \frac{l_x}{|l|}, \frac{l_y}{|l|}, \frac{l_z}{|l|} \). For the line, we are given \( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \). According to the condition: \[ \left( \frac{1}{c} \right)^2 + \left( \frac{1}{c} \right)^2 + \left( \frac{1}{c} \right)^2 = 1 \] Step 2: Solve for \( c \).
Simplifying the equation: \[ \frac{3}{c^2} = 1 \quad \Rightarrow \quad c^2 = 3 \quad \Rightarrow \quad c = \pm \sqrt{3} \] Step 3: Conclusion.
The correct answer is (C) \( c = \pm \sqrt{3} \).