QJEEM08Ma09a9: JEE Main 2008 Mathematics - Class 12 JEEM08Ma09a9 Solution: If the straight lines and intersects at a point, then the…

Question

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If the straight lines

\,\,\,\,\,

\,\,\,\,\,

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

\,\,\,\,\,

and

\,\,\,\,\,

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

intersects at a point, then the integer

k

is equal to

Accepted Answer

The two lines intersect if shortest distance between them is zero

i.e.

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

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

where

{\overrightarrow a _1} = \widehat i + 2\widehat j + 3\widehat k,{\overrightarrow b _1} = k\widehat i + 2\widehat j + 3\widehat k

\,\,\,\,\,\,\,{\overrightarrow a _2} = 2\widehat i + 3\widehat j + \widehat k,\,\,{\widehat b_2} = 3\widehat i + k\widehat j + 2\widehat k

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

\Rightarrow 1\left( {4 - 3k} \right) - 1\left( {2k - 9} \right) - 2\left( {{k^2} - 6} \right) = 0

\Rightarrow - 2{k^2} - 5k + 25 = 0 \Rightarrow k = - 5

\,\,\,\,

or

\,\,\,\,

{5 \over 2}

As

k

is an integer, therefore

k=-5

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