QJEEM21Ma4f7a: JEE Main 2021 Mathematics - Class 12 JEEM21Ma4f7a Solution: If the foot of the perpendicular from point (4, 3, 8) on the…

Question

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If the foot of the perpendicular from point (4, 3, 8) on the line

{L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}

, l

\ne

0 is (3, 5, 7), then the shortest distance between the line L₁ and line

{L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}

is equal to :

Accepted Answer

(3, 5, 7) lie on given line L₁

{{3 - a} \over l} = {3 \over 3} = {{7 - b} \over 4}

{{7 - b} \over 4} = 1 \Rightarrow b = 3

{{3 - a} \over l} = 1 \Rightarrow 3 - a = l

M(4, 3, 8)

N(3, 5, 7)

DR'S of MN = (1,

-

2, 1)

MN

\bot

line L₁

(1)(l) + (

-

2)(3) + 4(1) = 0

\Rightarrow

l = 2

a = 1

a = 1, b = 3, l = 2

{{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}

{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}

A = \, &lt; 1,2,3 &gt;

B = \, &lt; 2,4,5 &gt;

\overrightarrow {AB} = \, &lt; 1,2,2 &gt;

\overrightarrow p = 2\widehat i + 3\widehat j + 4\widehat k

\overrightarrow q = 3\widehat i + 4\widehat j + 5\widehat k

\overrightarrow p \, \times \overrightarrow q = - \widehat i + 2\widehat j - \widehat k

Shortest distance =

\left| {{{\overrightarrow {AB} \,.\,(\overrightarrow p \times \overrightarrow q )} \over {|\overrightarrow p \times \overrightarrow q |}}} \right|\, = {1 \over {\sqrt 6 }}

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