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+c√a2+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 = |12∗2–5∗1+9√122+(−5)2|
= |24−5+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 = ab√a2+b2
⟹ p2 = a2b2a2+b2 ⟹ 1p2 = a2+b2a2b2 ⟹ 1p2 = 1a2 + 1b2
Hence Proved.