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Q.
The value of the angle between two straight lines $y=(2-\sqrt{3}) x+5$ and $y=(2+\sqrt{3}) x-7$ is
ManipalManipal 2020
Solution:
Given equation of straight lines are
$y=(2-\sqrt{3}) x+5$
and $y=(2+\sqrt{3}) x-7$
On comparing with $y=m x+c$, we get
$m_{1}=2-\sqrt{3}$
and $m_{2}=2+\sqrt{3}$
$\therefore \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}$
$=\frac{2+\sqrt{3}-(2-\sqrt{3})}{1+(2-\sqrt{3})(2+\sqrt{3})}$
$=\frac{2 \sqrt{3}}{1+(4-3)}=\frac{2 \sqrt{3}}{1+4-3}$
$\Rightarrow \tan \theta=\frac{2 \sqrt{3}}{2}=\sqrt{3}$
$\Rightarrow \theta=60^{\circ}$