Q73 / 180 · KCET 2018 · Mathematics

The angle between the lines \( 2 x=3 y=-z \) and \( 6 x=-y=-4 z \) is

  • A
    \( 0^{\circ} \)
  • B
    \( 45^{\circ} \)
  • Correct answer
    C
    \( 90^{\circ} \)
  • D
    \( 30^{\circ} \)

Solution

Given lines, $2 x=3 y=-z$ $\Rightarrow \frac{x}{1 / 2}=\frac{y}{1 / 3}=\frac{z}{-1} \rightarrow(1)$ Direction ratios are $(1 / 2,1 / 3,-1)$ and $6 x=-y=-4 z$ $\Rightarrow \frac{x}{1 / 6}=\frac{y}{-1}=\frac{z}{-1 / 4} \rightarrow(2)$ Direction ratios are $(1 / 6,-1,-1 / 4)$ Now, $l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}$ $\frac{1}{12}-\frac{1}{3}+\frac{1}{4}=\frac{1-4+3}{12}=0$ Therefore, lines are perpendicular, that is, angle between two lines is $90^{\circ}$.

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

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