The combined equation of the lines passing through the origin and having slopes $\frac{2}{3}$ and $-\frac{2}{3}$ is

The area of the triangle formed by the lines $3x^2 - 4xy + y^2 = 0$ and $2x - y = 6$ is:

The perpendiculars are drawn to lines $L_1$ and $L_2$ from the origin making an angle $\frac{\pi}{4}$ and $\frac{3 \pi}{4}$ respectively with the positive direction of the $X$-axis. If both the lines are at a unit distance from the origin,then their joint equation is

If the lines $x^2-4xy+y^2=0$ make angles $\alpha$ and $\beta$ with the positive direction of the $X$-axis,then $\cot^2 \alpha + \cot^2 \beta = $

If the lines represented by the equation $ax^2 - bxy - y^2 = 0$ make angles $\alpha$ and $\beta$ with the $x$-axis,then $\tan(\alpha + \beta) = $

The value of $k$,if the slope of one of the lines given by $4x^2 + kxy + y^2 = 0$ is four times that of the other,is given by