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

Answer: (D)

6

Answer 2

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:

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