Question

Find the coordinates of the foot of perpendicular, draw form the point $(2,3)$ to line $y=3x-4$ ?

Answer 1

Given equation of line is;

$y=3x-4⟶\left(1\right)where,m=3$

since the equation of line perpendicular to (1) is given as;$y=m^{\prime }x+c⟶\left(2\right)$

where, $m^{\prime }=-\frac{1}{m}=-\frac{1}{3}$

also eqn (2) passes through $(2,3)$

then, $3=-\frac{1}{3}.2+c\Rightarrow c=\frac{11}{3}$

equation (2) becomes $y=-\frac{1}{3}x+\frac{11}{3}\Rightarrow 3y=-x+11⟶\left(3\right)$

solving (1) and (3) we get,

$x=\frac{23}{10};y=\frac{29}{10}$

coordinate of foot of the perpendicular is $(\frac{23}{10},\frac{29}{10})$