Question

If \(α,β,γ\) are the direction angles of the line \(x = 4z+3\) and \(y = 2-3z\), then the value of \(cosα+cosβ+cosγ\) is...

• A. \(\frac{8}{\sqrt{26}}\)
• B. \(\frac{6}{\sqrt{26}}\)
• C. \(\frac{4}{\sqrt{26}}\)
• D. \(\frac{2}{\sqrt{26}}\)

Hint:

Write the line in symmetric form to read off direction ratios, then divide by the magnitude.

Answer 1

The Correct Option is D

Step 1: Write in symmetric form
From \(x = 4z + 3\) we get \(\dfrac{x - 3}{4} = z\). From \(y = 2 - 3z\) we get \(\dfrac{y - 2}{-3} = z\). So
\[ \frac{x - 3}{4} = \frac{y - 2}{-3} = \frac{z}{1} \]

Step 2: Direction ratios
The direction ratios are \((4, -3, 1)\). The magnitude is \(\sqrt{16 + 9 + 1} = \sqrt{26}\).

Step 3: Direction cosines
\(\cos\alpha = \dfrac{4}{\sqrt{26}}\), \(\cos\beta = \dfrac{-3}{\sqrt{26}}\), \(\cos\gamma = \dfrac{1}{\sqrt{26}}\).

Step 4: Add
\[ \cos\alpha + \cos\beta + \cos\gamma = \frac{4 - 3 + 1}{\sqrt{26}} = \frac{2}{\sqrt{26}} \]
Option (D). Options (A) to (C) come from adding the magnitudes of the ratios or leaving out a term.

Final Answer:
The sum of direction cosines is 2/sqrt(26). This is option (D). \[ \boxed{\text{(D) }\frac{2}{\sqrt{26}}} \]