The joint equation of the pair of lines passing through the origin and making an angle of $\frac{\pi}{6}$ with the line $3x+y-6=0$ is

The combined equation of the lines passing through the point $(3,4)$ and each making an angle $45^{\circ}$ with the line $x+y+1=0$ is

If the equation of the pair of straight lines passing through the point $(1, 1)$ and perpendicular to the pair of lines $3x^2 + 11xy - 4y^2 = 0$ is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ (where the constant term is $12$),then find the value of $2(a - h + b - g + f - 12)$.

If the products of the perpendiculars from the origin to the pairs of lines $xy+x+y+1=0$,$x^2-y^2+2x+1=0$,and $2x^2+3xy-2y^2+2x+1=0$ are $p_1, p_2$,and $p_3$ respectively,then:

If $m_{1}$ and $m_{2}$ are slopes of the lines represented by $(\sec^{2} \theta - \sin^{2} \theta) x^{2} - 2 \tan \theta xy + \sin^{2} \theta y^{2} = 0$,then $|m_{1} - m_{2}| = $

If the slopes of the lines represented by $Kx^2 - 4xy + 5y^2 = 0$ differ by $2$,then $K = $