The Straight Lines — MATHS STD 11 Science — Question

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

$\Rightarrow\frac{\text{x}-2}{\frac{1}{\sqrt2}}=\frac{\text{y}-1}{\frac{1}{\sqrt2}}=\text{r}$ 

or $\text{x}=\frac{1}{\sqrt2}\text{r}+2,\ \text{y}=\frac{1}{\sqrt2}\text{r}+1$

$\text{B}\Big(\frac{\text{r}}{\sqrt2}+2,\ \frac{\text{r}}{\sqrt2}+1\Big)$ lie on x + 2y + 1 = 0

$\therefore\frac{\text{r}}{\sqrt2}+2+\frac{2\text{r}}{\sqrt2}+2+1=0$

$\frac{3\text{r}}{\sqrt2}=\pm5$

$\text{r}=\frac{5\sqrt2}{3}$

The lenght of AB is $\frac{5\sqrt2}{3}$ units.