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

Answer: (C)

$\frac{3\sqrt{14}}{7}$

Answer 2

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.