If one of the pair of lines $4x^2+6xy+ky^2=0$ is perpendicular to one of the lines represented by $3x^2-5xy+2y^2=0$,then twice the absolute difference of such possible values of $k$ is

The difference of the tangents of the angles which the lines $(\tan ^2 \alpha+\cos ^2 \alpha) x^2-2 x y \tan \alpha +(\sin ^2 \alpha) y^2=0$ make with the $X$-axis is

$2 x^2-3 x y-2 y^2=0$ represents two lines $L_1$ and $L_2$. $2 x^2-3 x y-2 y^2-x+7 y-3=0$ represents another two lines $L_3$ and $L_4$. Let $A$ be the point of intersection of lines $L_1$ and $L_3$,and $B$ be the point of intersection of lines $L_2$ and $L_4$. The area of the triangle formed by lines $AB$,$L_3$,and $L_4$ is

If $\left(\frac{2}{3}, 0\right)$ is the centroid of the triangle formed by the lines $4x^2-y^2=0$ and $lx+my+n=0$,then $l+m+n=$

The numbers $\alpha$ and $\beta$ are such that one of the lines of $2x^2 + \alpha xy + 3y^2 = 0$ coincides with one of the lines of $2x^2 + \beta xy - 3y^2 = 0$. If the two lines other than that common line are perpendicular,then $|\alpha + \beta|$ is equal to

If the equation of the pair of lines passing through $(1, 1)$ and perpendicular to the pair of lines $2x^2 + xy - y^2 - x + 2y - 1 = 0$ is $ax^2 + 2hxy + by^2 + 2gx + 3y = 0$,then $\frac{b}{a} =$