Why a is not equal to zero in this pair of straight line equations?

[rajeshmarndi]

Let say we have two line $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$. Then pair of straight line equation is

$a_1a_2x^2+(a_1b_2+b_1a_2)xy+b_1b_2y^2+(a_1c_2+c_1a_2)x+(b_1c_2+c_1b_2)y+c_1c_2=0$

i.e $ax^2+2hxy+by^2+2gx+2fy+c=0$

Now if we take $a_1=0$, then the first line becomes $b_1y+c_1=0$ i.e 1st line is parallel to x-axis . Then the pair of striaght line equation becomes $2{h}'xy+by^2+2{g}'x+2fy+c=0$

In other words the coefficient of $x^2$ i.e $a$ becomes zero and still we have the pair of straight line equation.

 

[BvU]

Correct. If I read your title well, you think this is not allowed ?

 

[PeroK]

Let say we have two line $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$. Then pair of straight line equation is

$a_1a_2x^2+(a_1b_2+b_1a_2)xy+b_1b_2y^2+(a_1c_2+c_1a_2)x+(b_1c_2+c_1b_2)y+c_1c_2=0$

i.e $ax^2+2hxy+by^2+2gx+2fy+c=0$

Now if we take $a_1=0$, then the first line becomes $b_1y+c_1=0$ i.e 1st line is parallel to x-axis . Then the pair of striaght line equation becomes $2{h}'xy+by^2+2{g}'x+2fy+c=0$

In other words the coefficient of $x^2$ i.e $a$ becomes zero and still we have the pair of straight line equation.

If you simplify right down to the lines $x = 0$ and $y =0$, you get the two-line equation $xy = 0$.