Official solution
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 + \[{{{P^2}} \over {49}}\]
\[ \Rightarrow \] \[{{8P} \over 7} = 4\] \[ \Rightarrow \] \[P = {7 \over 2}\]