Question

A line passing through the point A (5,0) makes an angle of 30^o with the positive direction of x-axis. It is rotated at point A through an angle of 15^o clockwise direction. Find the equation of the line in new position.

Answer 1

$$\begin{gathered} \mathrm{As}\mathrm{the}\mathrm{line}\mathrm{passing}\mathrm{through}\mathrm{the}\mathrm{point}\left(5,0\right)\mathrm{makes}30°\mathrm{with}\mathrm{positive}\mathrm{direction}\mathrm{of}\mathrm{Xaxis}. \\ \mathrm{After}\mathrm{the}\mathrm{rotation},\mathrm{the}\mathrm{line}\mathrm{will}\mathrm{make}\left(30°-15°\right)\mathrm{that}\mathrm{is}15°\mathrm{of}\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{positive}\mathrm{direction}\mathrm{of}\mathrm{X}\mathrm{axis}. \\ \mathrm{Equation}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{line}\mathrm{will}\mathrm{be}: \\ \mathrm{y}-{\mathrm{y}}_1=\mathrm{m}(\mathrm{x}-{\mathrm{x}}_1) \\ \mathrm{y}-0=\tan 15(\mathrm{x}-5) \\ \mathrm{y}=\frac{\sqrt{3}-1}{\sqrt{3}+1}(\mathrm{x}-5) \\ \Rightarrow \mathrm{y}\left(\sqrt{3}+1\right)=\sqrt{3}-1(\mathrm{x}-5) \\ \Rightarrow \left(\sqrt{3}+1\right)\mathrm{y}=\sqrt{3}\mathrm{x}-5\sqrt{3}-\mathrm{x}+5 \\ \Rightarrow \left(\sqrt{3}+1\right)\mathrm{y}=\left(\sqrt{3}+1\right)\mathrm{x}-5\left(\sqrt{3}-1\right)\mathrm{that}\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{equation}. \end{gathered}$$