First slide

Equation of line in Straight lines

Question

The distance of the point $(2, 3)$ from the line $2 x - 3 y + 9 = 0$ measured along the line $x - y + 1 = 0,$ is

Moderate

A

4
B

$2 \sqrt{2}$
C

$8 \sqrt{2}$
D

$4 \sqrt{2}$

Solution

The slope of the line $x - y + 1 = 0$ is 1. So, it makes an angle of $45 °$ with $x -$ axis.
The equation of a line passing through $(2, 3)$ and making an angle of $45 °$ is given by
$\frac{x - 2}{\cos 45^{\circ}} = \frac{y - 3}{\sin 45^{\circ}} = r$
Co-ordinates of any point on this line are
$(2 + r \cos 45^{\circ}, 3 + r \sin 45^{\circ})$ or, $(2 + \frac{r}{\sqrt{2}}, 3 + \frac{r}{\sqrt{2}})$
If this point lies on the line $2 x - 3 y + 9 = 0$ , then
$4 + r \sqrt{2} - 9 - \frac{3 r}{\sqrt{2}} + 9 = 0 \Rightarrow r = 4 \sqrt{2}$
Hence, required distance $= 4 \sqrt{2}$ units .

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