Q265 / 270 · NDA 2016 (Phase 1) · Mathematics

What is the acute angle between the lines represented by the equations $y-\sqrt{3} \mathrm{x}-5=0$ and $\sqrt{3} \mathrm{y}-\mathrm{x}+6=0$ ?

  • Correct answer
    A
    $30^{\circ}$
  • B
    $45^{\circ}$
  • C
    $60^{\circ}$
  • D
    $75^{\circ}$

Solution

$\begin{aligned} & y-\sqrt{3} x-5=0...(i) \\ & \sqrt{3} y-x+6=0...(ii") \\ & y=m x+c \end{aligned}$ From (i) and from (ii) $\begin{array}{ll} y=\sqrt{3} x+5 & y=\frac{x}{\sqrt{3}}-\frac{6}{\sqrt{3}} \\ m_1=\sqrt{3} & m_2=\frac{1}{\sqrt{3}} \end{array}$ Angle between two lines, $\begin{aligned} & \tan \theta=\left|\frac{m_1-m_2}{1+m_1 m_2}\right| \\ &=\left|\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{1+\sqrt{3} \frac{1}{\sqrt{3}}}\right|=\frac{1}{\sqrt{3}} \\ &=\tan 30^{\circ} \\ & \theta=30^{\circ} \end{aligned}$

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

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