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:

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

Answer 2

Step 1: Extract direction ratios (DRs) of the first line
Given in symmetric form: \[ \frac{2 - x}{3}=\frac{3y - 2}{4\lambda + 1}=4 - z = t. \] Write parametrically: \[ x=2-3t,\quad y=\frac{(4\lambda+1)t+2}{3},\quad z=4-t. \] Hence the direction vector (coefficients of \(t\)) is \[ \vec{d}_1=\langle -3,\ \tfrac{4\lambda+1}{3},\ -1\rangle. \] To avoid fractions, scale by 3: \[ \vec{D}_1=\langle -9,\ 4\lambda+1,\ -3\rangle. \]

Step 2: Extract direction ratios (DRs) of the second line
Given: \[ \frac{x+3}{3\mu}=\frac{1-2y}{6}=\frac{5-z}{7}=s. \] Parametric form: \[ x=3\mu s-3,\quad y=\frac{1-6s}{2}=\frac{1}{2}-3s,\quad z=5-7s. \] Thus the direction vector is \[ \vec{D}_2=\langle 3\mu,\ -3,\ -7\rangle. \]

Step 3: Use the right angle condition
Lines are perpendicular if their direction vectors are orthogonal: \[ \vec{D}_1\cdot \vec{D}_2=0. \] Compute: \[ (-9)(3\mu) + (4\lambda+1)(-3) + (-3)(-7) = 0 \] \[ -27\mu -12\lambda -3 + 21 = 0 \;\;\Longrightarrow\;\; -27\mu -12\lambda + 18 = 0. \] Multiply by \(-1\): \[ 27\mu + 12\lambda - 18 = 0 \;\;\Longrightarrow\;\; 12\lambda + 27\mu = 18. \]

Step 4: Compute the required expression
Divide both sides by 3: \[ 4\lambda + 9\mu = 6. \]

Final answer

6