Question

The equation of a line passing through origin with direction angles \( \frac{2\pi}{3}, \frac{\pi}{4}, \frac{\pi}{3} \) is

• A. \( x = \frac{y}{\sqrt{2}} = z \)
• B. \( \frac{x}{-1} = \frac{y}{-\sqrt{2}} = z \)
• C. \( x = \frac{y}{-\sqrt{2}} = z \)
• D. \( x = \frac{y}{-\sqrt{2}} = -z \)

Hint:

Direction cosines are \( (\cos\alpha,\cos\beta,\cos\gamma) \)Use them directly to form direction ratios and then write symmetric equation.

Answer 1

The Correct Option is D

Step 1: Write direction cosines.
\[ l = \cos\frac{2\pi}{3} = -\frac{1}{2}, \quad m = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}, \quad n = \cos\frac{\pi}{3} = \frac{1}{2} \]

Step 2: Form direction ratios.
Direction ratios are proportional to direction cosines:
\[ l : m : n = -\frac{1}{2} : \frac{1}{\sqrt{2}} : \frac{1}{2} \]

Step 3: Remove fractions.
Multiply by 2:
\[ -1 : \frac{2}{\sqrt{2}} : 1 \]
\[ = -1 : \sqrt{2} : 1 \]

Step 4: Write symmetric form of line.
\[ \frac{x}{-1} = \frac{y}{\sqrt{2}} = \frac{z}{1} \]

Step 5: Simplify the equation.
Multiply signs to match option form:
\[ x = \frac{y}{-\sqrt{2}} = -z \]

Step 6: Match with options.
This matches option (D).

Step 7: Final conclusion.
\[ \boxed{x = \frac{y}{-\sqrt{2}} = -z} \]