If ax^2+2hxy+by^2+2gx+2fy+c=0 represents a pa | vedclass

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(B) The general second-degree equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of parallel lines if $h^2=ab$ and $bg^2=af^2$.From the condition

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$\sqrt{\frac{a}{b}}$

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(B) The general second-degree equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of parallel lines if $h^2=ab$ and $bg^2=af^2$.From the condition $h^2=ab$,we have $h=\sqrt{ab}$.From the condition $bg^2=af^2$,we can write $\frac{g^2}{f^2}=\frac{a}{b}$.Taking the square root on both sides,we get $\frac{g}{f}=\sqrt{\frac{a}{b}}$.For parallel lines,the distance between them is given by $2\sqrt{\frac{g^2-ac}{a(a+b)}} = 2\sqrt{\frac{f^2-bc}{b(a+b)}}$.This implies $\frac{g^2-ac}{a} = \frac{f^2-bc}{b}$.Rearranging the terms,we get $\frac{g^2-ac}{f^2-bc} = \frac{a}{b}$.Therefore,$\sqrt{\frac{g^2-ac}{f^2-bc}} = \sqrt{\frac{a}{b}}$.

List-$I$	List-$II$
$(A)$ $5x^2 + 2\sqrt{7}xy - y^2 = 0$	$(I)$ $\frac{\sqrt{3}}{2}$
$(B)$ $x^2 + \sqrt{11}xy + 2y^2 = 0$	$(II)$ $\frac{1}{2\sqrt{3}}$
$(C)$ $x^2 + 2\sqrt{2}xy + y^2 = 0$	$(III)$ $\frac{1}{2}$
$(D)$ $3x^2 + 4\sqrt{2}xy + y^2 = 0$	$(IV)$ $\frac{2}{3}$
	$(V)$ $\frac{1}{\sqrt{2}}$

If $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of parallel lines,then $\sqrt{\frac{g^2-ac}{f^2-bc}}$ is equal to

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