If the sum of the slopes of the lines represented by the equation $x^2 - 2xy \tan A - y^2 = 0$ is $4$,then $\angle A = $

The combined equation of the lines passing through the origin making an acute angle $\alpha$ with the line $y=x$ is

If one of the lines of the pair $ax^{2}+2hxy+by^{2}=0$ bisects the angle between the positive direction of the axes,then $a, b,$ and $h$ satisfy the relation:

If the slope of one of the two lines represented by the equation $\frac{x^2}{a} + \frac{2xy}{h} + \frac{y^2}{b} = 0$ is twice that of the other, then find the ratio $h^2 : ab$.

If the equation $7x^2 - 14xy + py^2 - 12x + qy - 4 = 0$ represents a pair of parallel lines,then the value of $\sqrt{p^2 + q^2 - pq}$ is

The combined equation of the straight lines of the form $y=kx+1$ (where $k$ is an integer) such that the point of intersection of each with the line $3x+4y=9$ has an integer as its $x$-coordinate is