Distance of a Point from a Line – Formula and Example

Here you will learn formula to find the distance of a point from a line with examples.

Let’s begin –

Distance of a Point from a Line

The length of the perpendicular from a point (x1,y1) to a line ax + by + c = 0 is

|ax1+by1+ca2+b2|.

It is the distance of a point from a line.

Distance of a Line from Origin

The length of the perpendicular from the origin to a line ax + by + c = 0 is

|c|a2+b2.

Algorithm to find distance :

Step 1 : Write the equation of the line in the form ax + by + c = 0

Step 2 : Substitute the coordinates x1 and y1 of the point in place of x and y respectively in the expression.

Step 3 : Divide the result obtained in step 2 by the square root of the sum of the squares of the coefficients of x and y.

Step 4 : Take the modulus of the expression obtained in step 3.

he result obtained after step 4 is the required distance.

Example : Find the distance between the line 12x – 5y + 9 = 0 and the point (2,1).

Solution : We have line 12x – 5y + 9 = 0 and the point (2,1)

Required distance = |12251+9122+(5)2|

= |245+9|13 = 2813

Example : If p is the length of the perpendicular from the origin to the line xa + yb = 1, then prove that 1p2 = 1a2 + 1b2

Solution : The given line is bx + ay – ab = 0 ………….(i)

It is given that

p = Length of the perpendicular from the origin to line (i)

p = |b(0)+a(0)ab|b2+a2 = aba2+b2

p2 = a2b2a2+b2 1p2 = a2+b2a2b2 1p2 = 1a2 + 1b2

Hence Proved.

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