A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is , then is __________. [2024]

Question

A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is $l$ , then $l^{2}$ is __________. [2024]

Smart Achievers · Accepted Answer

(65)

We have, $l_{1}$ : $\frac{x}{1} = \frac{y + 2}{1} = \frac{z}{1} = λ \in R$

Coordinates of point P are $(λ, λ - 2, λ)$

$l_{2}$ : $\frac{x + 2}{2} = \frac{y}{1} = \frac{z}{1} = μ \in R$

Coordinates of point Q are $(2 μ - 2, μ, μ)$

D.r.'s of PQ are $(2 μ - 2 - λ, μ - λ + 2, μ - λ)$

Also, D.r.'s of line PQ is (2, 1, 2)

$∴ \frac{2 μ - 2 - λ}{2} = \frac{μ - λ + 2}{1} = \frac{μ - λ}{2}$

$\Rightarrow λ = 6, μ = 2$

$∴$ Coordinates of point P are (6, 4, 6) and coordinates of point Q are (2, 2, 2).

Equations of line PQ is $\frac{x - 2}{2} = \frac{y - 2}{1} = \frac{z - 2}{2} = k \in R$

Now, from condition for perpendicularity,

Since, AB $⊥$ PQ, then,

2(2k + 1) + (1)k + 2(2k – 10) = 0

$\Rightarrow 9 k = 18 \Rightarrow k = 2$

Therefore, point A is (6, 4, 6)

Now, perpendicular distance from B(1, 2, 12) to line PQ is given by

$l$ $= \sqrt{{(6 - 1)}^{2} + {{(4 - 2)}^{2} + (6 - 12)}^{2}}$ $\Rightarrow l^{2}$ = 25 + 4 + 36 = 65.

Solution

Q.
A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is $l$ , then $l^{2}$ is __________. [2024]

Want Us to Email you About Special Offers & Updates?

Solution

Q. A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l2 is __________. [2024]

Want Us to Email you About Special Offers & Updates?

Q.
A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is $l$ , then $l^{2}$ is __________. [2024]