QJEEM18Ma644a: JEE Main 2018 Mathematics - Class 12 JEEM18Ma644a Solution: If the angle between the lines, and is then p is equal to :

Question

Verified Solution • 2025–26 Session moderate

If the angle between the lines,

{x \over 2} = {y \over 2} = {z \over 1}

and

{{5 - x} \over { - 2}} = {{7y - 14} \over p} = {{z - 3} \over 4}\,\,

is

{\cos ^{ - 1}}\left( {{2 \over 3}} \right),

then p is equal to :

Accepted Answer

Let

\theta

be the angle between the two lines

Here direction cosines of

{x \over 2}

=

{y \over 2}

=

{z \over 1}

are 2, 2, 1

Also second line can be written as :

{{x - 5} \over 2} = {{y - 2} \over {{P \over 7}}} = {{z - 3} \over 4}

\therefore

its direction cosines are 2,

{{P \over 7}}

, 4

Also, cos

\theta

=

{2 \over 3}

(Given)

\because

cos

\theta

= \left| {{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}} \over {\sqrt {a_1^2 + b_1^2 + c_1^2\sqrt {a_2^2 + b_2^2 + c_2^2} } }}} \right|

\Rightarrow

{2 \over 3}

= \left| {{{\left( {2 \times 2} \right) + \left( {2 \times {P \over 7}} \right) + \left( {1 \times 4} \right)} \over {\sqrt {{2^2} + {2^2} + {1^2}} \sqrt {{2^2} + {{{P^2}} \over {49}} + {4^2}} }}} \right|

= {{4 + {{2P} \over 7} + 4} \over {3 \times \sqrt {{2^2} + {{{P^2}} \over {49}} + {4^2}} }}

\Rightarrow

{\left( {4 + {P \over 7}} \right)^2} = 20 + {{{P^2}} \over {49}}

\Rightarrow

16 +

{{8P} \over 7} + {{{P^2}} \over {49}}

= 20 +

{{{P^2}} \over {49}}

\Rightarrow

{{8P} \over 7} = 4

\Rightarrow

P = {7 \over 2}

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