Question

Find the coordinates of the foot of perpendicular, image and equation of perpendicular drawn from the $(2,3)$ to the line $y=3x-4$

Answer 1

R.E.F image

Equation of line $AB$ is

$y-3=-\frac{1}{3}(x-2)$

$\Rightarrow 3y-4=-x+2$

$\Rightarrow x+3y=11\to \left(I\right)$

$3x-y-4=0\to \left(II\right)$

$1x3$

$3x+4y-33=0$

$\underset{–}{3x-u-4=0}$

$10y-29=0$

$=y=\frac{29}{10}$

$\therefore x=11-3\times \frac{29}{10}=\frac{110-87}{10}$

$=\frac{23}{10}$

$\therefore x=\frac{23}{10}$

$\therefore$ Full of perpendicular $B(\frac{23}{10},\frac{29}{10})$

$\therefore \frac{\lambda +2}{2}=\frac{23}{10},\frac{k+3}{2}=\frac{29}{10}$

$=\lambda =\frac{23}{5}-2,k+3=\frac{29}{5}$

$\therefore \lambda =\frac{13}{5},k=\frac{14}{5}$

Image is $C(\frac{13}{5},\frac{14}{5})$

Equation of perpendicular line is $x+3y=11$