A straight line is drawn through the point $\left(\sqrt{3}, 2\right)$ and inclined at an angle of $30^\circ$ to the $X$ axis. Find the length intercepted on the line between the given point and the line $\sqrt{3}x + y = 9$.
The given line passes through the point $\left(\sqrt{3}, 2\right)$ and is inclined at an angle of $30^\circ$ to the $X$ axis.
$\therefore \;$ The equation of the line can be written as
$\dfrac{x - \sqrt{3}}{\cos 30^\circ} = \dfrac{y - 2}{\sin 30^\circ} = r$ $\;\;\; \cdots \; (1)$
where $\;$ $r$ $\;$ is the distance between $\left(\sqrt{3}, 2\right)$ and any point $P \left(x, y\right)$.
We have from equation $(1)$
$x = \sqrt{3} + r \cos 30^\circ = \sqrt{3} + \dfrac{\sqrt{3} r}{2}$
and $\;$ $y = 2 + r \sin 30^\circ = 2 + \dfrac{r}{2}$
If $P \left(x, y\right)$ be the point where the line given by equation $(1)$ meets the line $\;$ $\sqrt{3}x + y = 9$ $\;\;\; \cdots \; (2)$,
the coordinates of $P$ must satisfy equation $(2)$.
Substituting the values of $x$ and $y$ in equation $(2)$ gives
$\sqrt{3} \left(\sqrt{3} + \dfrac{\sqrt{3}r}{2}\right) + 2 + \dfrac{r}{2} = 9$
i.e. $\;$ $3 + \dfrac{3r}{2} + 2 + \dfrac{r}{2} = 9$
i.e. $\;$ $2r = 4$ $\implies$ $r = 2$
i.e. $\;$ the length intercepted $= 2$