A) \[-\frac{7}{4}\]
B) \[\frac{2}{7}\]
C) \[-\frac{4}{7}\]
D) \[\frac{7}{2}\]
Correct Answer: D
Solution :
Above formula is used to find angle\[\theta \]between two lines having direction ratios\[{{a}_{1}},{{b}_{1}},{{c}_{1}}\]and\[{{a}_{2}},{{b}_{2}},{{c}_{2}}\] Now let us find the direction cosines of the given lines \[\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\]direction cosines are 2, 2, 1 next line can also be written as \[\frac{x-5}{2}=\frac{y-2}{\frac{P}{7}}=\frac{z-3}{4}\] Hence direction cosines are \[2,\frac{P}{7},4\] and we know that \[\cos \theta =\frac{2}{3}\] Keeping the value of these cosines in above formula we get \[\frac{2}{3}=\left| \frac{2\times 2+2\times \frac{P}{7}+1\times 4}{\sqrt{{{2}^{2}}+{{2}^{2}}+{{1}^{2}}}\sqrt{{{2}^{2}}+\frac{{{P}^{2}}}{49}+{{4}^{2}}}} \right|=\frac{8+\frac{2P}{7}+4}{3\times \sqrt{{{2}^{2}}+\frac{{{P}^{2}}}{49}+{{4}^{2}}}}\] After cross multiplication we get \[\Rightarrow \]\[{{\left( 4+\frac{P}{7} \right)}^{2}}=20+\frac{{{P}^{2}}}{49}\] \[\Rightarrow \]\[16+\frac{8P}{7}+\frac{{{P}^{2}}}{49}=20+\frac{{{P}^{2}}}{49}\] \[\Rightarrow \]\[\frac{8P}{7}=4\] \[\Rightarrow \]\[P=\frac{7}{2}\] Therefore Answer is D \[\cos \left| \frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \right|\]
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