Question

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

• A. $2\sqrt{2}$
• B. $3\sqrt{2}$
• C. $4\sqrt{2}$
• D. $4\sqrt{2}$

Answer 1

The correct option is A $2\sqrt{2}$


Let the point be $P$, lines be $L_1$ & $L_2$ and ${}^{\prime }{d}^{\prime }$ be the distance of point $P(1,3)$ from the line $L_1$ along the direction of line $L_2$
Let the line segment from point $P$ intersect line $L_1$ at $O$

$\Rightarrow$ slope of $L_2=$ slope of $OP$
$L_2:y=x+1$
$\Rightarrow$ slope of $L_2=1$
$\Rightarrow$ slope of $OP=1$

we have slope of $OP$ and point $P(1,3)$
Equation of $OP$ will be
$y=x+2...\left(1\right)$
$L_1:2x-3y+9=0...\left(2\right)$
Coordinates of $O$ can be obtained from eqn $\left(1\right)$ and $\left(2\right)$
$O=(3,5)$

$OP=\sqrt{(3-1{)}^2+(5-3{)}^2}=2\sqrt{2}$