Explanation

(a)
∵ The slope of line $x + y = 1$ is $- 1$
$\therefore\$It makes an angle of $135{^\circ}$ with $x$-axis
The equation of line passing through (1, 1) and making an angle of $135{^\circ}$ is
$\frac{x - 1}{\cos{135{^\circ}}} = \frac{y - 1}{\sin{135{^\circ}}} = r$
$\Rightarrow \frac{x - 1}{- \frac{1}{\sqrt{2}}} = \frac{y - 1}{\frac{1}{\sqrt{2}}} = r$
Coordinates of any point on this line are
$\left( 1 - \frac{r}{\sqrt{2}},\ 1 + \frac{r}{\sqrt{2}} \right)$
If this point lies on $2x - 3y = 4,$ then
$2\left( 1 - \frac{r}{\sqrt{2}} \right) - 3\left( 1 + \frac{r}{\sqrt{2}} \right) = 4$
$\Rightarrow 2 - \frac{2r}{\sqrt{2}} - 3 - \frac{3r}{\sqrt{2}} = 4$
$\Rightarrow \frac{5r}{\sqrt{2}} = - 5$
$\Rightarrow r = \sqrt{2}$ (neglect negative sign)