Question

If the line \(\frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z\) makes a right angle with the line  \(\frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7},\) then \( 4\lambda + 9\mu \) is equal to:

• A. 13
• B. 4
• C. 5
• D. 6

Answer 1

The Correct Option is D

Given:

\[ \frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z \tag{1} \]

From equation (1), we have:

\[ \frac{x - 2}{-3} = \frac{y - 2}{3} = \frac{z - 4}{-1} \]

Now consider the second line:

\[ \frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7} \tag{2} \]

From equation (2), we have:

\[ \frac{x + 3}{3\mu} = \frac{y - \frac{1}{2}}{-3} = \frac{z - 5}{-7} \]

Since the lines are perpendicular, their direction ratios should satisfy:

\[ (-3)(3\mu) + (4\lambda + 1)(-1) + (-1)(-7) = 0 \]

Expanding this:

\[ -9\mu - 4\lambda - 1 + 7 = 0 \] \[ 4\lambda + 9\mu = 6 \]