QJEEM12Ma9513: JEE Main 2012 Mathematics - Class 12 JEEM12Ma9513 Solution: If the line and intersect, then is equal to :

Question

Verified Solution • 2025–26 Session moderate

If the line

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

and

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

intersect, then

k

is equal to :

Accepted Answer

Given lines in vector form are

\overrightarrow r = \left( {\widehat i - \overrightarrow j + \overrightarrow k } \right) + \lambda \left( {2\overrightarrow i + 3\overrightarrow j + 4\overrightarrow j } \right)

and

\overrightarrow r = \left( {3\widehat i + k\widehat j} \right) + \mu \left( {\widehat i + 2\widehat j + \widehat k} \right)

These will intersect if shortest distance between them

=0

i.e.

\left( {{{\overrightarrow a }_2} - {{\overrightarrow a }_1}} \right).\overrightarrow {{b_1}} \times {\overrightarrow b _2} = 0

\Rightarrow | \begin{matrix} 3 - 1 & k + 1 & - 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{matrix} | = 0

\Rightarrow 2\left( { - 5} \right) - \left( {k + 1} \right)\left( { - 2} \right) - 1\left( 1 \right) = 0

\Rightarrow k = 9/2

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