The acute angle between the lines $x \cos 30^{\circ}+y \sin 30^{\circ}=3$ and $x \cos 60^{\circ}+y \sin 60^{\circ}=5$ is

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Aicharya · Accepted Answer

Given equations of lines, $x \cos 30^{\circ}+y \sin 30^{\circ}=3$ and $x \cos 60^{\circ}+y \sin 60^{\circ}=5$ $\Rightarrow \sqrt{3} x+y=6$ and $x+\sqrt{3} y=10$ $	herefore \quad$ Slope of line $\sqrt{3} x+y=6=m_1=-\sqrt{3}$ Slope of line $x+\sqrt{3} y=10=\mathrm{m}_2=\frac{-1}{\sqrt{3}}$ $\begin{aligned} 	herefore \quad 	an 	heta & =\left|\frac{m_1-m_2}{1+m_1 m_2}ight| \ & =\left|\frac{-\sqrt{3}+\frac{1}{\sqrt{3}}}{1+(-\sqrt{3}) 	imes\left(\frac{-1}{\sqrt{3}}ight)}ight| \ & =\left|\frac{-3+1}{2 \sqrt{3}}ight| \ & =\left|\frac{-2}{2 \sqrt{3}}ight| \ 	herefore \quad 	an 	heta & =\frac{1}{\sqrt{3}} \ 	herefore \quad \Rightarrow 	heta & =30^{\circ}\end{aligned}$

The acute angle between the lines $x \cos 30^{\circ}+y \sin 30^{\circ}=3$ and $x \cos 60^{\circ}+y \sin…

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