Q44 / 90 · JEE Main 2011 · Mathematics

If the angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=4$ is $\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then $\lambda$ equals

  • A
    $\frac{3}{2}$
  • B
    $\frac{2}{5}$
  • C
    $\frac{5}{3}$
  • Correct answer
    D
    $\frac{2}{3}$

Solution

$\cos \theta=\sqrt{\frac{5}{14}}$ $\sin \theta=\frac{3}{\sqrt{14}}$ $\sin \theta=\frac{1+4+3 \lambda}{\sqrt{1+4+\lambda^2} \sqrt{1+4+9}}$ $\frac{3}{\sqrt{14}}=\frac{5+3 \lambda}{\sqrt{5+\lambda^2} \sqrt{14}} \Rightarrow \lambda=\frac{2}{3}$

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

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