Question

If \( (\alpha, \beta, \gamma) \) are the direction cosines of an angular bisector of two lines whose direction ratios are (2,2,1) and (2,-1,-2), then \( (\alpha + \beta + \gamma)^2 \) is:

• A. \( 3 \)
• B. \( 2 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

For angular bisector problems, normalize the given direction ratios and use the standard bisector formula to find the required expression.

Answer 1

The Correct Option is B

Step 1: Finding direction cosines of the angular bisector The formula for the direction cosines of the angular bisector of two lines with direction ratios \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) is: \[ \alpha = \frac{l_1}{\sqrt{l_1^2 + m_1^2 + n_1^2}} + \frac{l_2}{\sqrt{l_2^2 + m_2^2 + n_2^2}} \] Similarly, we compute \( \beta \) and \( \gamma \), then find \( (\alpha + \beta + \gamma)^2 \). Using calculations, we get: \[ (\alpha + \beta + \gamma)^2 = 2 \]