Q100 / 160 · TS EAMCET 2016 · Mathematics

If the angle between the lines whose direction cosines are $\left(-\frac{2}{\sqrt{21}}, \frac{C}{\sqrt{21}}, \frac{1}{\sqrt{21}}\right)$ and $\left(\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}},-\frac{6}{\sqrt{54}}\right)$ is $\frac{\pi}{2}$, then the value of $C$ is

  • A
    6
  • Correct answer
    B
    4
  • C
    -4
  • D
    2

Solution

Given, $$ \begin{aligned} & I_1=-\frac{2}{\sqrt{21}}, m_1=\frac{C}{\sqrt{21}}, n_1=\frac{1}{\sqrt{21}} \\ & \text { and } \quad l_2=\frac{3}{\sqrt{54}}, m_2=\frac{3}{\sqrt{54}}, n_2=\frac{-6}{\sqrt{54}} \\ & \because \quad \cos \theta=l_1 1_2+m_1 m_2+n_1 n_2 \\ & \therefore \\ & \cos \frac{\pi}{2}=-\frac{2}{\sqrt{21}} \times \frac{3}{\sqrt{54}}+\frac{C}{\sqrt{21}} \times \frac{3}{\sqrt{54}}+\frac{1}{\sqrt{21}} \times-\frac{6}{\sqrt{54}} \\ & \Rightarrow \quad 0=\frac{-6}{\sqrt{21} \sqrt{54}}+\frac{3 C}{\sqrt{21} \sqrt{54}}-\frac{6}{\sqrt{21} \sqrt{54}} \\ & \Rightarrow \quad \frac{12}{\sqrt{21} \sqrt{54}}=\frac{3 C}{\sqrt{21} \sqrt{54}} \\ & \Rightarrow \quad 3 \mathrm{C}=12 \\ & \therefore \quad C=4 \\ & \end{aligned} $$

Hence, the correct answer is option: 'B'.

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