Pair of Straight lines :
It is combined product of more than one straight lines.
Let us assume —
For two lines :
L1 ::: ax + by + c = 0 ….(1)
L2 ::: a1x + b1y + c1 = 0. ….(2)
Hence (ax + by + c)( a1x + b1y + c1 ) = 0 is known as joint equation of lines (1) lines (2).
So , it is also known as “ pair of straight lines ”.
Note : To find joint equation you must need to make R.H.S of two lines equal to zero and then multiply of both equation.
Ex. Find the joint equation of lines-
y = x. And y = -x.
Solution :
Equation can be rewritten as—
y – x = 0 and y + x = 0
Then , combined both equation ( y – x ) ( y + x) = 0.
y2 — x2 = 0
Wrong method :
Lines : y = x
y = —x
y2 = — x2
y2 + x2 = 0
To find separate equation from joint equation :
First of all make R.H.S equal to zero and then resolve L.H.S into two linear factors or you can also use Shri Dharacharya method.
Ex .
Evaluate separate equation from joint equation ( pair of straight lines )
x2 —6 xy + 8y2 = 0
Solution :
First Method :
Factorization —
x2 —6 xy + 8y2 = 0
( x – 4y ) ( x – 2y ) = 0
x – 4y = 0
x – 2y = 0
Second method :
Shri Dharacharya method —
x2 —6 xy + 8y2 = 0
x = 4y or x = 2y
x – 4y = 0 &
x – 2y = 0