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
4.Distance between Two parallel lines :
Let the two parallel lines be ax + by + c = 0 and parallel line ax + by + c₁ = 0
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 a ᵢ x + b ᵢ y + c ᵢ = 0
i = 1 ,2 ,3 are concurrent if
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
