1. **State the problem:** We need to find the equation of a line $L$ that passes through the point $(3, -2)$ and is inclined at an angle of $60^\circ$ to the line $\sqrt{3}x + y = 1$. The line $L$ also intersects the x-axis. 2. **Find the slope of the given line:** The given line is $\sqrt{3}x + y = 1$. Rewrite it in slope-intercept form: $$y = -\sqrt{3}x + 1$$ So, the slope of this line is $m_1 = -\sqrt{3}$. 3. **Use the angle between two lines formula:** If $m_1$ and $m_2$ are slopes of two lines inclined at an angle $\theta$, then $$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$ Given $\theta = 60^\circ$, so $\tan 60^\circ = \sqrt{3}$. 4. **Set up the equation for $m_2$:** $$\sqrt{3} = \left| \frac{-\sqrt{3} - m_2}{1 - \sqrt{3} m_2} \right|$$ 5. **Solve for $m_2$:** Consider the positive case: $$\sqrt{3} = \frac{-\sqrt{3} - m_2}{1 - \sqrt{3} m_2}$$ Multiply both sides by denominator: $$\sqrt{3}(1 - \sqrt{3} m_2) = -\sqrt{3} - m_2$$ $$\sqrt{3} - 3 m_2 = -\sqrt{3} - m_2$$ Bring all terms to one side: $$\sqrt{3} + \sqrt{3} = - m_2 + 3 m_2$$ $$2 \sqrt{3} = 2 m_2$$ $$m_2 = \sqrt{3}$$ 6. **Check the negative case:** $$\sqrt{3} = - \frac{-\sqrt{3} - m_2}{1 - \sqrt{3} m_2} = \frac{\sqrt{3} + m_2}{1 - \sqrt{3} m_2}$$ Multiply both sides: $$\sqrt{3}(1 - \sqrt{3} m_2) = \sqrt{3} + m_2$$ $$\sqrt{3} - 3 m_2 = \sqrt{3} + m_2$$ $$-3 m_2 - m_2 = \sqrt{3} - \sqrt{3}$$ $$-4 m_2 = 0$$ $$m_2 = 0$$ 7. **Possible slopes for line $L$ are $m_2 = \sqrt{3}$ or $m_2 = 0$.** 8. **Find the equation of line $L$ passing through $(3, -2)$ with slope $m_2$:** - For $m_2 = \sqrt{3}$: $$y - (-2) = \sqrt{3}(x - 3)$$ $$y + 2 = \sqrt{3} x - 3 \sqrt{3}$$ $$y = \sqrt{3} x - 3 \sqrt{3} - 2$$ - For $m_2 = 0$: $$y - (-2) = 0(x - 3)$$ $$y + 2 = 0$$ $$y = -2$$ 9. **Check which line intersects the x-axis:** - For $y = \sqrt{3} x - 3 \sqrt{3} - 2$, set $y=0$: $$0 = \sqrt{3} x - 3 \sqrt{3} - 2$$ $$\sqrt{3} x = 3 \sqrt{3} + 2$$ $$x = 3 + \frac{2}{\sqrt{3}}$$ This is a valid intersection. - For $y = -2$, the line is parallel to x-axis and does not intersect it. 10. **Final answer:** The equation of line $L$ is $$y = \sqrt{3} x - 3 \sqrt{3} - 2$$