The joint equation of a pair of lines passing through the origin,each of which makes an angle of $30^{\circ}$ with the $Y$-axis,is

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

Assertion $(A)$: The difference of the slopes of the lines represented by $y^2 - 2xy \sec^2 \alpha + (3 + \tan^2 \alpha)(\tan^2 \alpha - 1) x^2 = 0$ is $4$.
Reason $(R)$: The difference of the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is $\frac{2 \sqrt{h^2 - ab}}{|b|}$.

If the equation $ax^2 + 2hxy + by^2 = 0$ has one line as the bisector of the angle between the coordinate axes,then:

The perpendiculars are drawn to lines $L_1$ and $L_2$ from the origin making an angle $\frac{\pi}{4}$ and $\frac{3 \pi}{4}$ respectively with the positive direction of the $X$-axis. If both the lines are at a unit distance from the origin,then their joint equation is

Which among the following represents the combined equation of a pair of lines passing through the point $(1, 0)$ and parallel to the lines represented by $2x^2 - xy - y^2 = 0$?