Question

The distance of the point $(1,4)$ from the line $3x-2y-5=0$ measured along the line $x-y+3=0$ is

• A. $8\sqrt{2}$
• B. $9\sqrt{2}$
• C. $10\sqrt{2}$
• D. $12\sqrt{2}$

Answer 1

The correct option is C $10\sqrt{2}$

$x-y+3=0$
$\Rightarrow$ Slope, $m=1$
So, it makes an angle of ${45}^{\circ }$ with the x-axis.

Equation of the line passing through $(1,4)$ and making an angle of ${45}^{\circ }$ is given by
$\frac{x-1}{\cos {45}^{\circ }}=\frac{y-4}{\sin {45}^{\circ }}=r$

Let $P$ be any point on this line.
$\therefore$ Coordinates of $P$ is $(1+\frac{r}{\sqrt{2}},4+\frac{r}{\sqrt{2}})$

If this point lies on the line $3x-2y-5=0$, then
$3+\frac{3r}{\sqrt{2}}-8-\frac{2r}{\sqrt{2}}-5=0$
$\Rightarrow r=10\sqrt{2}$
Hence, required distance $=10\sqrt{2}$