If the lines represented by a x^2+2 h x y+b y | vedclass

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(A) Given the equation of the pair of lines: $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$. Since the lines intersect on the $x$-axis,we set $y=0$ to find the

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$a b c=2 f g h$

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(A) Given the equation of the pair of lines: $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$. Since the lines intersect on the $x$-axis,we set $y=0$ to find the intersection points on the $x$-axis: $a x^2+2 g x+c=0$. For the lines to intersect at a point on the $x$-axis,the roots of this quadratic must be equal (or the point of intersection lies on the axis),implying the discriminant $D=0$: $(2 g)^2 - 4(a)(c) = 0$ $\Rightarrow 4 g^2 = 4 a c$ $\Rightarrow g^2 = a c$. For the general second-degree equation to represent a pair of straight lines,the condition is: $a b c+2 f g h-a f^2-b g^2-c h^2=0$. Substituting $g^2=a c$ into this condition: $a b c+2 f g h-a f^2-b(a c)-c h^2=0$ $a b c+2 f g h-a f^2-a b c-c h^2=0$ $2 f g h = a f^2 + c h^2$. Also,using the condition for intersection,it can be shown that $a f^2 = c h^2$. Comparing these,$a b c = 2 f g h$ is not generally true.

If the lines represented by $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ intersect on the $x$-axis,which of the following is in general incorrect?

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