Problem to Find perpendicular line from a point to a line in a coordinate system

Topic : Co ordinate System and a Line Equation

Question : Use co ordinate geometry to show the perpendicular distance between a point (x,y) and a line Ax+By+C= 0, is given by

d = (Ax+By+C) / √(A²+B²)

Solution :

Perpendicular Distance from (0,0) to the line Ax+By+C = 0 is
C
--------------
√(A²+B²)

Effecting the translation (X,Y)=>(X+x,Y+y) the equation of the line changes to :

A(X+x)+B(Y+y)+C=0
AX+BY+(Ax+By+C)=0
Perpendicular distance from the origin is :
(AX+BY+C ) / √(A²+B²) = (Ax+By+C ) / √(A²+B²)

Since the transformation here is an isometry the distance is preserved


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