Question

If the distance of the point $(2,3)$ from the line $2x-3y+9=0$ measured along the line $x-y+1=0$ is $k$ units, then the value of $k^2$ is

Answer 1

Slope of the line $x-y+1=0$ is $1$
So angle made by the line in anticlockwise direction with $+vex-$axis is ${45}^{\circ }$
Now, parametric equation of the line having the slope as $1$ and passing though $(2,3)$ will be
$x=2+r\cos {45}^{\circ };y=3+r\sin {45}^{\circ }\Rightarrow x=2+\frac{r}{\sqrt{2}};y=3+\frac{r}{\sqrt{2}}$

Now substituting the values of $x,y$ in the line $2x-3y+9=0$
$4+\sqrt{2}r-9-\frac{3r}{\sqrt{2}}+9=0\Rightarrow r=4\sqrt{2}\Rightarrow k=4\sqrt{2}\therefore k^2=32$