Question

A line which passes through $P(4,5)$ and making an angle of ${30}^{\circ }$ with positive direction of $x-$axis. Then coordinates of point which is at a distance $4$ unit's from the line on either side of $P$, is

• A. $(4-2\sqrt{3},7),(4+2\sqrt{3},3)$
• B. $(4-2\sqrt{3},7),(4+2\sqrt{3},7)$
• C. $(4+2\sqrt{3},7),(4-2\sqrt{3},3)$
• D. $(4+2\sqrt{3},3),(4+2\sqrt{3},7)$

Answer 1

The correct option is C $(4+2\sqrt{3},7),(4-2\sqrt{3},3)$

Since, the equation of the line passing through $P(5,4)$ and making an angle ${30}^{\circ }$
$\frac{x-4}{\cos {30}^{\circ }}=\frac{y-5}{\sin {30}^{\circ }}=r$
$\therefore$ Point on the line at a distance 4 from $(4,5)$ is given by
$\frac{x-4}{\cos {30}^{\circ }}=\frac{y-5}{\sin {30}^{\circ }}=\pm 4$
$x=4\pm 4\cos {30}^{\circ }$
and $y=5\pm 4\sin {30}^{\circ }$
$\Rightarrow x=4\pm 2\sqrt{3}$
and $y=5\pm 2$
The coordinates of the point are $(4+2\sqrt{3},7)$ and $(4-2\sqrt{3},3)$