If the two lines given by $ax^2+2hxy+by^2=0$ make inclinations $\alpha$ and $\beta$ with the $x$-axis,then $\tan(\alpha+\beta)=$

The joint equation of the pair of lines passing through the origin with slopes $(1+\sqrt{2})$ and $\frac{1}{(1+\sqrt{2})}$ is

If angles made by lines $x^2 - 4xy - y^2 = 0$ with the positive direction of the $x$-axis are $\theta_1$ and $\theta_2$,then the value of $\sec^2(\theta_1 + \theta_2) + |\frac{1}{\tan \theta_1} + \frac{1}{\tan \theta_2}|$ is

The slope of one of the pair of lines $2x^2 + hxy + 6y^2 = 0$ is thrice the slope of the other line. Then,$h = $

If the equation $3x^2 + xy - y^2 - 3x + 6y + k = 0$ represents a pair of lines,then $k$ is equal to

The joint equation of the pair of lines passing through $(2,3)$ and parallel to the lines represented by $x^{2}-y^{2}=0$ is: