The condition for the general quadratic equat | vedclass

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Question

(C) The general quadratic equation is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.The condition for the equation to represent a pair of straight lines is

Vedclass · Accepted Answer

$\Delta = 0, h^2 = ab, g^2 = ac$ and $f^2 = bc$

Vedclass · Answer

(C) The general quadratic equation is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.The condition for the equation to represent a pair of straight lines is $\Delta = 0$,where $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.For the lines to be parallel,the condition is $h^2 = ab$.For the lines to be coincident,the distance between the parallel lines must be zero.The distance $d$ between parallel lines is given by $d = 2\sqrt{\frac{g^2 - ac}{a(a+b)}}$ or $d = 2\sqrt{\frac{f^2 - bc}{b(a+b)}}$.Setting $d = 0$ implies $g^2 = ac$ and $f^2 = bc$.Thus,the combined condition for coincident lines is $\Delta = 0, h^2 = ab, g^2 = ac$,and $f^2 = bc$.Therefore,the correct option is $C$.

The condition for the general quadratic equation $f(x, y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent coincident lines is:

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