# Introduction of Pair of straight lines

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### 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

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