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Straight Line — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsStraight Line3 Marks
Question
A line perpendicular to segment joining $A(1, 0)$ and $B(2, 3)$ divides it internally in the ratio $1 : 2.$ Find the equation of the line
✓

Answer

Given, $A(1,0), B(2,3)$
Slope of $A B=\frac{3-0}{2-1}=3$
Required line is perpendicular to $A B$.
Slope of required line $=\frac{-1}{3}$
Let point $C$ divide $A B$ in the ratio $1: 2$.
$C & \equiv\left(\frac{1(2)+2(1)}{1+2}, \frac{1(3)+2(0)}{1+2}\right) $
$ =\left(\frac{4}{3}, \frac{3}{3}\right) $
$=\left(\frac{4}{3}, 1\right)$
Required line passes through $\left(\frac{4}{3}, 1\right)$ and has slope $=\frac{-1}{3}$
Equation of the line in slope point form is $y-y_1=m\left(x-x_1\right)$
The equation of the required line is
$ y-1=\frac{-1}{3}\left(x-\frac{4}{3}\right)$
$\Rightarrow 3(y-1)=-1\left(x-\frac{4}{3}\right)$
$\Rightarrow 3 y-3=-x+\frac{4}{3}$
$\Rightarrow 9 y-9=-3 x+4$
$\Rightarrow 3 x+9 y=13 $

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