Question

A straight line drawn through the point A (2, 1) making an angle $\pi /4$ with positive x-axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.

Answer 1

$\text{Here}(x_1,y_1)=A(2,1),\alpha =\frac{x}{4}$

$\text{The equation of line is}$

$\frac{x-x_1}{cos\alpha }=\frac{y-y_1}{sin\alpha }=r$

$\frac{x-2}{cos\alpha }=\frac{y-1}{sin\alpha }=r$

$\frac{x-2}{cos45}=\frac{y-1}{sin45}=r$

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

$orx=\frac{1}{\sqrt{2}}r+2,y=\frac{1}{\sqrt{2}}r+1$

$B(\frac{r}{\sqrt{2}}+2,\frac{r}{\sqrt{2}}+1)lieonx+2y+1=0$

$\therefore \frac{r}{\sqrt{2}}+2+\frac{2r}{\sqrt{2}}+2+1=0$

$\frac{3r}{\sqrt{2}}=-5$

$r=-\frac{5\sqrt{2}}{3}$

$\text{The length AB is}\frac{5\sqrt{2}}{3}\text{units}.$