Question

A line passes through a point A (1, 2) and makes an makes an angle of ${60}^{\circ }$ with the x-axis and intersects the line x + y = 6 at the point P. Find AP.

Answer 1

$\text{The equation of line through (1, 2) and making an angle of}{60}^{\circ }\text{with the x-axis is}$

$\frac{x-1}{cos{60}^{\circ }}=\frac{y-2}{sin{60}^{\circ }}=r$

$\frac{x-1}{\frac{1}{2}}=\frac{y-2}{\frac{\sqrt{3}}{2}}=r$

$\text{Where r is the distance of any point on the line from A (1, 2)}$

$\text{The coordinates of p on the line are}$

$(1+\frac{1}{2}r,2+\frac{\sqrt{3}}{2}r)$

$\text{and}$

$\text{p lies on}x+y=6$

$\therefore 1+\frac{r}{2}+2+\frac{\sqrt{3}r}{2}=6$

$or3+\frac{r}{2}+\frac{\sqrt{3}}{2}r=6$

$=\frac{r}{2}(1+\sqrt{3})=6-3$

$=\frac{r}{2}(1+\sqrt{3})=3$

$r=\frac{6}{1+\sqrt{3}}$

$orr\frac{6}{1+\sqrt{3}}=3(\sqrt{3}-1)$

$\text{Hence length AP}=3(\sqrt{3}-1).$