Find the co-ordinates of the foot of the perpendicular drawn from the point$$(2,3)$$to the line $$y=3x+4$$
Equation of given line is $$y=3x+4$$
Let $$Q(x_{1},y_{1})$$ be the foot of the perpendicular on the line.
$$\Rightarrow y_{1}=3x_{1}+4$$ ......(i)
Slope of given line is $$3.$$
Slope of perpendicular $$PQ \displaystyle =-\frac{1}{3}$$
$$\Rightarrow \displaystyle \frac { { y }_{ 1 }-3 }{ { x }_{ 1 }-2 } =-\frac { 1 }{ 3 } $$
$$\Rightarrow 3y_{1}=-x_{1}+11$$ .....(ii)
Solving (i) and (ii), we get
$$\displaystyle x_{1}=-\frac{1}{10} ,y_{1}=\frac{37}{10}$$
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