If the real pair of lines L_1 : ax^2 + 2hxy + | vedclass

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(D) The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of angle bisectors,which are always perpendicular to each other. Therefore,the sum of the

Vedclass · Accepted Answer

The pair of lines given by $L_1$ is the angle bisector of lines $lx^2 + 2mxy + ny^2 = 0$ for all $l, m, n \in \mathbb{R}$.

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(D) The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of angle bisectors,which are always perpendicular to each other. Therefore,the sum of the coefficients of $x^2$ and $y^2$ must be zero,i.e.,$a + b = 0$. Since the origin $(0, 0)$ is the point of intersection of the angle bisectors,it must satisfy the equation of the lines $L_2$. Substituting $(0, 0)$ into $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$,we get $F = 0$. Thus,$a + b + F = 0 + 0 = 0$. For the general second-degree equation to represent a pair of lines,the determinant $\Delta$ must be zero: $ABC + 2hfg - af^2 - bg^2 - ch^2 = 0$. Applying this condition to $L_2$ leads to the relation $BDE = AE^2 + CD^2$. Option $D$ is incorrect because the angle bisectors of a pair of lines depend on the specific coefficients of those lines,not just any arbitrary $l, m, n$.

If the real pair of lines $L_1 : ax^2 + 2hxy + by^2 = 0$ represents the angle bisectors of the real lines given by $L_2 : Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$,then which of the following is incorrect?

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