Q56 / 170 · Manipal (MET) 2020 · Mathematics

The value of the angle between two straight lines $y=(2-\sqrt{3}) x+5$ and $y=(2+\sqrt{3}) x-7$ is

  • A
    $30^{\circ}$
  • Correct answer
    B
    $60^{\circ}$
  • C
    $45^{\circ}$
  • D
    $90^{\circ}$

Solution

Given equation of straiaht lines are $y=(2-\sqrt{3}) x+5$ and $\quad y=(2+\sqrt{3}) x-7$ On comparing with $y=m x+c$, we get $m_1=2-\sqrt{3}$ and $\quad m_2=2+\sqrt{3}$ $\therefore \quad \tan \theta=\frac{m_2-m_1}{1+m_1 m_2}$ $=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2+\sqrt{3})}$ $=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{1+4-3}$ $\Rightarrow \quad \tan \theta=\frac{2 \sqrt{3}}{2}=\sqrt{3}$ $\Rightarrow \quad \theta=60^{\circ}$

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

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