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.

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