Properties of a straight line

Properties of a straight line –

Let equation ax + by + c = 0

1.Method to Write a line parallel to a given line –

Change only the constant term in the given equation, the term of x and y remaining unchanged. The changed constant will be found by an additional given condition.

For example –

ax + by + c = 0 is

ax + by + k = 0

3x – 4y – 7 = 0 is

3x – 4y + k = 0.

Note-

A line passing  through (h , k) and parallel to

Line -:  ax + by + c = 0 is

parallel  line  : a( x – h ) + b( y – k) = 0.

This is another method to write a line parallel to a given line and passing through a given point.

2.Method to Write a line perpendicular to a given line

Interchange the coefficient of x and y  in the given equation and change the sign in between them.

And also change the constant term. The value of the new constant is to be found by an additional given condition.

For example –

A line perpendicular to

ax + by + c = 0 is

bx – ay + k = 0

And

3x – 4y – 7 = 0 is

4x – 3y + k = 0.

Note-

A line passing  through (h, k) and perpendicular to

Line -:  ax + by + c = 0 is

Perpendicular line  :  b( x – h ) – a( y – k) = 0.

This is another method to write a line perpendicular to a given line and passing through a given point.

3.The foot of perpendicular :

Method to find the foot of the perpendicular

Foot of perpendicular from a point ( h , k) to a given line ax + by + c = 0

foot of perpendicular
4.Distance between Two parallel lines :

Let the two parallel lines  be ax + by + c = 0  and parallel line  ax + by + c1 = 0

Distance between Two parallel lines
5.Perpendicular distance from outside point P(x1,y1) ( not lying on a line) on a line   ax + by + c = 0
6.Perpendicular distance from outside point P(x1,y1) ( not lying on a line) on a line   x cos a + y sin a   =  p

is

| x1 cos a + y1 sin a     p | .

7.To find the point of intersection of two lines—

Lines must be non-parallel  line . Its mean parallel line does not meet .

Suppose , the two lines be

a1x + b1y + c1 = 0.  and  a2x + b2y + c2 = 0

If (x1 , y1 ) be the co-ordinates of their point of intersection—

Then

a1x + b1y + c1 = 0.  and 

a2x + b2y + c2 = 0

solving these two equation and find ( x , y )

8.Concurrent lines :

If  three given lines are concurrent they must be meet in a common point.

Method to find concurrency—

Method 1. Firstly we find point of intersection on solving any two lines. If this point satisfy the third equation also , the given lines are concurrent.

Method 2.  Determinant form

Let three lines  ai x + bi y + c i = 0

i = 1 ,2 ,3 are concurrent if

concurrent lines
9.A line equally inclined with two lines —

Let the two lines and has a slope m1 and m2  be equally inclined to a line with slope m , then

A line equally inclined with two lines