If a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 repres | vedclass

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(A) The given equation is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$. For option $(A)$,if the slopes of the two lines are $m_1$ and $m_2$,then $m_1+m_2 = -

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If the slope of one line is negative of the slope of another line,then $h = 0$

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(A) The given equation is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$. For option $(A)$,if the slopes of the two lines are $m_1$ and $m_2$,then $m_1+m_2 = -\frac{2h}{b}$ and $m_1 m_2 = \frac{a}{b}$. If $m_1 = -m_2$,then $m_1+m_2 = 0$,which implies $-\frac{2h}{b} = 0$,so $h = 0$. Thus,option $(A)$ is correct. For option $(B)$,if the lines are parallel,then $h^2 = ab$ and $bg^2 = af^2$. The condition $2f(gh+af) = 0$ is not generally true for parallel lines. For option $(C)$,if the lines intersect at the origin,the constant term $c$ must be $0$ and the linear terms $2gx$ and $2fy$ must be $0$,so $g=f=c=0$. The condition $h^2=ab$ is for the lines to be parallel,not necessarily for intersection at the origin. For option $(D)$,the $x$-coordinate of the point of intersection is given by $\frac{bg-hf}{h^2-ab}$. The condition $hf-bg > 0$ is equivalent to $bg-hf < 0$,which does not guarantee the $x$-coordinate is positive.

If $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a pair of lines,which of the following statements is true?

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