The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0$. Then the value of $k$ is:

The combined equation of a pair of lines passing through the origin and inclined at $60^{\circ}$ and $30^{\circ}$ respectively with the $x$-axis is:

Let $L_1, L_2$ be the lines represented by the equation $4x^2-5xy+3y^2=0$. Let $L_3, L_4$ be two lines passing through the point $(4,3)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equation of $L_3$ and $L_4$ is $ax^2+2hxy+by^2+2gx+2fy+c=0$,then find the value of $af+bg+ch$.

If $\frac{x^2}{a} + \frac{2xy}{h} + \frac{y^2}{b} = 0$ represents a pair of straight lines and the slope of one of the lines is twice that of the other,then $ab : h^2$ is:

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 equation $xy + a^2 = a(x + y)$ represents