Question

Let \( P \) be the point of intersection of the lines \[ \frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1} \quad \text{and} \quad \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2}. \] Then, the shortest distance of \( P \) from the line \( 4x = 2y = z \) is:

• A. \( \frac{5\sqrt{14}}{7} \)
• B. \( \frac{\sqrt{14}}{7} \)
• C. \( \frac{3\sqrt{14}}{7} \)
• D. \( \frac{6\sqrt{14}}{7} \)

Answer 1

The Correct Option is C

Find the point of intersection \( P \):

Line \( L_1 \):

\[ \frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1} = \lambda. \]

Parametric coordinates of \( P \) on \( L_1 \):

\[ x = \lambda + 2, \quad y = 5\lambda + 4, \quad z = \lambda + 2. \]

Line \( L_2 \):

\[ \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2} = \mu. \]

Parametric coordinates of \( P \) on \( L_2 \):

\[ x = 2\mu + 3, \quad y = 3\mu + 2, \quad z = 2\mu + 3. \]

Equate \( x, y, z \) for both lines:

\[ \lambda + 2 = 2\mu + 3 \implies \lambda = 2\mu + 1, \] \[ 5\lambda + 4 = 3\mu + 2 \implies 5(2\mu + 1) + 4 = 3\mu + 2, \] \[ 10\mu + 5 + 4 = 3\mu + 2 \implies 7\mu = -7 \implies \mu = -1. \]

Substituting \( \mu = -1 \):

\[ \lambda = 2(-1) + 1 = -1. \]

Coordinates of \( P \):

\[ P(1, -1, 1). \]

Find the shortest distance from \( P \) to the line \( 4x = 2y = z \):

The equation of the line \( L_3 \) in symmetric form is:

\[ \frac{x}{1/4} = \frac{y}{1/2} = \frac{z}{1}. \]

Let \( Q(k, 2k, 4k) \) be a point on \( L_3 \). Direction ratios of \( PQ \):

\[ \text{DRs of } PQ = (k - 1, 2k + 1, 4k - 1). \]

\( PQ \perp L_3 \): Solve using:

\[ (k - 1)\cdot\frac{1}{4} + (2k + 1)\cdot\frac{1}{2} + (4k - 1)\cdot 1 = 0. \]

Simplify:

\[ \frac{k - 1}{4} + \frac{4k + 2}{4} + 4k - 1 = 0, \] \[ \frac{k - 1}{4} + \frac{4k + 2}{4} + \frac{16k - 4}{4} = 0, \] \[ 21k - 3 = 0 \implies k = \frac{1}{7}. \]

Coordinates of \( Q \):

\[ Q\left(\frac{1}{7}, \frac{2}{7}, \frac{4}{7}\right). \]

Calculate \( PQ \):

\[ PQ = \sqrt{\left(\frac{1}{7} - 1\right)^2 + \left(\frac{2}{7} + 1\right)^2 + \left(\frac{4}{7} - 1\right)^2}. \]

Simplify:

\[ PQ = \sqrt{\left(-\frac{6}{7}\right)^2 + \left(\frac{9}{7}\right)^2 + \left(-\frac{3}{7}\right)^2}. \] \[ PQ = \sqrt{\frac{36}{49} + \frac{81}{49} + \frac{9}{49}} = \sqrt{\frac{126}{49}} = \sqrt{\frac{36}{7}} = \frac{3\sqrt{14}}{7}. \]

Option (3) is correct.