condition for pair of straight line equation

Question

While determining the condition for the pair of straight line equation $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ i.e $\quad$$ax^2+2(hy+g)x+(by^2+2fy+c)=0$ $$x=\frac{-2(hy+g)}{2a}\pm\frac{\sqrt{(hy+g)^2-a(by^2+2fy+c)}}{a} $$$$x=\frac{-2(hy+g)}{2a}\pm\frac{\sqrt{(h^2-ab)y^2+2(hg-af)y+(g^2-ac)}}{a}$$ The terms inside square root need to be a perfect square. I understand this. What I do not understand is when the inside square root terms ,quadratic in y is taken to be zero. Because of which its determinant $4(hg-af)^2-4(h^2-ab)(g^2-ac)=0$ becomes the condition for the pair of straight line equation.

I am stuck here. Can someone please help. Thanks.

Answer 1

For $A\ge0,$

$$Ax^2+Bx+C=A(x+B/2A)^2+C-B^2/4A$$ will be perfect square for all real values of $x$ iff $$C-B^2/4A=0$$

Answer 2

If the condition

$$ a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0 $$

represents the product of two lines then the set of solutions for this conditions should be one solution point, void or infinite solution points associated to the cases in which we have the two lines intersection, two lines parallel and two lines coincident. With this idea we follow obtaining

$$ x = \frac{gh-af\pm2\sqrt{(g+hy)^2-a(b y^2+2fy+c)}}{2a} $$

If the intersection point is unique then the condition is

$$ (g + h y)^2 - a (c + 2 f y + b y^2) = 0 $$

which gives

$$ y = \frac{gh-af\pm\sqrt{a^2 f^2 + a b g^2 - 2 a f g h + a c h^2-a^2 b c}}{ab-h^2} $$

but again if the intersection point is unique we should have

$$ a^2 f^2 + a b g^2 - 2 a f g h + a c h^2-a^2 b c = 0 $$

or

$$ c = \frac{a f^2+b g^2-2 f g h}{a b-h^2} $$

The condition for distinct parallel lines follow as

$$ a b -h^2 = 0 $$

Another approach:

Assuming $a \ne 0$ and dividing $ax^2+2hxy+by^2+2gx+2fy+c=0$ by $a$ we have

$$ x^2+b' y^2 + c' + 2 f' y + 2 g'x +2 h' xy = (x+c_1 y + c_2)(x+d_1 y + d_2) $$

after equating coefficients and solving for $c_1,c_2,d_1,d_2$ we have the conditions

$$ \cases{h'^2-b' > 0\\ c' = -\frac{f'^2-b' g'^2-2f' g' h'}{h'^2-b'}} $$

such that the two lines equivalence is feasible.

or equivalently

$$ \cases{ h^2-a b > 0\\ c = \frac{a f^2+b g^2-2f g h}{a b -h^2} } $$

Answer 3

The equation (ax+by+cz)(dx+ey+fz)=0 when multiplied out is a homogeneous quadratic in x, y, z, and therefore it is the equation of a conic.

May suppose then that the quadratic is the conic whose equation is $ax^2+by^2+cz^2+2fyz+2gzx+2hxy=o$.

The condition for the conic to be two straight lines (assumed distinct) is that the determinant of coefficients vanishes viz $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}=0$