**Pair of Straight lines :**

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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.**

**y**^{2 }— x^{2 } = 0

**Wrong method :**

Lines : y = x

y = —x

y^{2 }= — x^{2}

**y**^{2 }+ x^{2 } = 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 )

**x**^{2 }—6 xy + 8y^{2 } = 0

**Solution :**

**First Method :**

**Factorization —**

x^{2 }—6 xy + 8y^{2 } = 0

( x – 4y ) ( x – 2y ) = 0

**x – 4y = 0**

**x – 2y = 0**

**Second method : **

**Shri Dharacharya method —**

x^{2 }—6 xy + 8y^{2 } = 0