Geometry: Distance From a Point to a Line

Distance from a point to a line:

We know that the perpendicular distance from origin to a straight line whose equation is a+by+c=0 is |c|/√(a²+b²). This formula works to find out the distance of a line from the origin whose coordinates are (0,0). Then how can we find the distance from a point different from origin to a given line? Let us see how we can find.

Let (h,k) be a point and the equation of line be ax+by+c=0. Let us find the distance now.

Let (x,y) be a point on the given line ax+by+c=0.

For finding the distance, let us shift the origin from (0,0) to (h,k) as shown below.

Let the coordinates of (x,y) with respect to the new axes would be (X,Y). Then,

x = X+h and y = Y+k

Substituting these values into the given equation, we get

a(X+h) + b(Y+k) + c =0

aX + bY + (ah+bk+c)=0

Since the perpendicular distance from origin to a straight line whose equation is a+by+c=0 is |c|/√(a²+b²), we have

The perpendicular distance from (h,k) to ax+by+c=0 is, |ah+bk+c|/√(a²+b²).

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