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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]

Ans.

(65)

We have, l1 : x1=y+21=z1=λ∈R

Coordinates of point P are (λ, λ–2, λ)

      l2 : x+22=y1=z1=μ∈R

Coordinates of point Q are (2μ–2, μ, μ)

D.r.'s of PQ are (2μ–2–λ, μ–λ+2, μ–λ)

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

∴  2μ–2–λ2=μ–λ+21=μ–λ2

⇒ 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 x–22=y–21=z–22=k∈R

Now, from condition for perpendicularity,

Since, AB ⊥ PQ, then,

    2(2k + 1) + (1)k + 2(2k – 10) = 0

⇒ 9k=18 ⇒ k=2

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

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

 l=(6–1)2+(4–2)2+(6–12)2⇒ l2 = 25 + 4 + 36 = 65.