(ii) If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ , then $L_1$ and $L_2$ never intesect, where $L_1$ and $L_2$ are parallel lines and having no solution.

(iii) If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}$ , then $L_1$ and $L_2$ are coincidenc linea and having infinitel many solutions.

Examples for Corresponding Conditions:

(i) $L_1 : 2x+3y-9=0, \quad L_2 :5x-3y+2=0$

(ii) $L_1 : 4x+8=0, \quad L_2 :2x+4y+3=0$

(iii) $L_1 : 2x-3y+7=0, \quad L_2 :6x-9y+21=0$

Family of lines through the point of intersection of two lines:

Let $L_1 : a_1x+b_1y+c_1=0$ -(i)

$L_2 :a_2x+b_2y+c_2=0$ -(ii)

Now consider the equation $(a_1x+b_1y+c_1)+\lambda (a_2x+b_2y+c_2) =0$ -(iii), represents family or system of lines where $\lambda \in R$ .

Conditions:

$(a_1x+b_1y+c_1)+\lambda (a_2x+b_2y+c_2) =0$ , represents family of lines.

(i) through the point of intersections $L_1$ and $L_2$ , if they intersect.

(ii) parallel to $L_1$ and $L_2$ , if they are parallel.

Examples for Corresponding Conditions:

$(i) \quad L_1: 2x-5y+2 =0, \quad L_2: 5x-4y-5=0, \quad L_3: (2x-5y+2)+\lambda(5x-4y+5)=0$

$(ii)\quad L_1: 2x-5y+3 =0, \quad L_2: 54x-10y+1=0, \quad L_3: (2x-5y+3)+\lambda(4x-10y+1)=0$

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