# 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 $$(x_1, y_1)$$ to a line ax + by + c = 0 is

|$$ax_1 + by_1 + c\over {\sqrt{a^2+b^2}}$$|.

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

$$| c |\over {\sqrt{a^2+b^2}}$$.

Algorithm to find distance :

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

Step 2 : Substitute the coordinates $$x_1$$ and $$y_1$$ 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 = |$$12*2 – 5*1 + 9\over {\sqrt{12^2 + (-5)^2}}$$|

= $$|24-5+9|\over 13$$ = $$28\over 13$$

Example : If p is the length of the perpendicular from the origin to the line $$x\over a$$ + $$y\over b$$ = 1, then prove that $$1\over p^2$$ = $$1\over a^2$$ + $$1\over b^2$$

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)

$$\implies$$ p = $$|b(0) + a(0) – ab|\over {\sqrt{b^2+a^2}}$$ = $$ab\over \sqrt{a^2+b^2}$$

$$\implies$$ $$p^2$$ = $$a^2b^2\over a^2+b^2$$ $$\implies$$ $$1\over p^2$$ = $$a^2+b^2\over a^2b^2$$ $$\implies$$ $$1\over p^2$$ = $$1\over a^2$$ + $$1\over b^2$$

Hence Proved.