Question

Let \( L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} \) be two lines. Then which of the following points lies on the line of the shortest distance between \( L_1 \) and \( L_2 \)?

• A. \( \left( \frac{-5}{3}, -7, 1 \right) \)
• B. \( (2, 3, \frac{1}{3}) \)
• C. \( \left( \frac{8}{3}, -1, \frac{1}{3} \right) \)
• D. \( \left( \frac{14}{3}, -3, \frac{22}{3} \right) \)

Hint:

The shortest distance between two skew lines is given by the line joining the points of intersection on each line.

Answer 1

The Correct Option is D

Shortest Distance Between Two Lines: Solution

Line \( PQ \) represents the shortest distance between lines \( L_1 \) and \( L_2 \). Point \( P \) is on \( L_1 \) with coordinates \( P(2\lambda + 1, 3\lambda + 2, 4\lambda + 3) \). Point \( Q \) is on \( L_2 \) with coordinates \( Q(3\mu + 2, 4\mu + 4, 5\mu + 5) \).

Step 1: Direction Ratios of \( PQ \)

The direction ratios of \( PQ \) are calculated as: \( (3\mu - 2\lambda + 1, 4\mu - 3\lambda + 2, 5\mu - 4\lambda + 2) \). The condition \( PQ \perp L_2 \) means the dot product of the direction ratios of \( PQ \) and \( L_2 \) is zero.

Step 2: First Equation from Dot Product

The dot product of \( PQ \) and \( L_2 \) yields the equation: \[ (3\mu - 2\lambda + 1) \times 2 + (4\mu - 3\lambda + 2) \times 3 + (5\mu - 4\lambda + 2) \times 4 = 0 \] Simplifying this gives: \[ 6\mu - 4\lambda + 2 + 12\mu - 9\lambda + 6 + 20\mu - 16\lambda + 8 = 0 \] Which reduces to: \[ 38\mu - 29\lambda + 16 = 0 \quad \text{...(1)} \]

Step 3: Second Equation

Applying \( PQ \perp L_1 \) results in the second equation: \[ (3\mu - 2\lambda + 1) \times 3 + (4\mu - 3\lambda + 2) \times 4 + (5\mu - 4\lambda + 2) \times 5 = 0 \] Simplifying this yields: \[ 9\mu - 6\lambda + 3 + 16\mu - 12\lambda + 8 + 25\mu - 20\lambda + 10 = 0 \] Which simplifies to: \[ 50\mu - 38\lambda + 21 = 0 \quad \text{...(2)} \]

Step 4: Solve System of Equations

Solving the system of equations: \[ 38\mu - 29\lambda + 16 = 0 \quad \text{...(1)} \] \[ 50\mu - 38\lambda + 21 = 0 \quad \text{...(2)} \] yields the values: \[ \lambda = \frac{1}{3}, \quad \mu = \frac{-1}{6} \]

Step 5: Coordinates of \( P \) and \( Q \)

Substituting \( \lambda = \frac{1}{3} \) and \( \mu = \frac{-1}{6} \) into the parametric forms: \[ P\left( 2\left(\frac{1}{3}\right) + 1, 3\left(\frac{1}{3}\right) + 2, 4\left(\frac{1}{3}\right) + 3 \right) = \left( \frac{5}{3}, 3, \frac{13}{3} \right) \] \[ Q\left( 3\left(\frac{-1}{6}\right) + 2, 4\left(\frac{-1}{6}\right) + 4, 5\left(\frac{-1}{6}\right) + 5 \right) = \left( \frac{3}{2}, \frac{10}{3}, \frac{25}{6} \right) \]

Step 6: Equation of Line \( PQ \)

The parametric equation for line \( PQ \) is: \[ \frac{x - \frac{5}{3}}{\frac{1}{6}} = \frac{y - 3}{\frac{-1}{3}} = \frac{z - \frac{13}{3}}{\frac{1}{6}} \] Simplified form: \[ \frac{x - \frac{5}{3}}{1} = \frac{y - 3}{-2} = \frac{z - \frac{13}{3}}{1} \]

Step 7: Verification of Point on Line \( PQ \)

Checking if point \( \left( \frac{14}{3}, -3, \frac{22}{3} \right) \) lies on \( PQ \): \[ \frac{x - \frac{5}{3}}{1} = \frac{14/3 - 5/3}{1} = \frac{9}{3} = 3 \] \[ \frac{y - 3}{-2} = \frac{-3 - 3}{-2} = \frac{-6}{-2} = 3 \] \[ \frac{z - \frac{13}{3}}{1} = \frac{22/3 - 13/3}{1} = \frac{9}{3} = 3 \] All ratios are equal, confirming the point is on the line.

Conclusion

The point \( \left( \frac{14}{3}, -3, \frac{22}{3} \right) \) is confirmed to lie on line \( PQ \), validating the calculations.