If $s$ and $p$ are respectively the sum and the product of the slopes of the lines $3x^2 - 2xy - 15y^2 = 0$,then $s:p$ is equal to

The equation of the pair of lines passing through the origin whose sum and product of slopes are respectively the arithmetic mean and geometric mean of $4$ and $9$ is

If the lines represented by the equation $6x^2 + 41xy - 7y^2 = 0$ make angles $\alpha$ and $\beta$ with the $x$-axis,then $\tan \alpha \cdot \tan \beta = $

The equations of the lines represented by the equation $ax^2 + (a + b)xy + by^2 + x + y = 0$ are

The lines represented by the equation $x^2-y^2-x+3y-2=0$ are :

If two lines represented by the equation $x^2 - (1 + \sqrt{3})xy + \sqrt{3}y^2 = 0$ make angles $\alpha$ and $\beta$ with the $X$-axis, then find the value of $\tan(\alpha + \beta)$.