The equation of the line passing through $(4, -6)$ and making an angle of $45^\circ$ with the positive $x$-axis is:

The equation of the line passing through $(1, 2)$ and parallel to the line $y = 3x - 1$ is

The system of linear equations $ax - y + 2 = 0$ and $x + ay + 3 = 0$ has a unique solution. Then,the set of values of $a$ is . . . . . . .

Find the equation of the line passing through the point of intersection of the lines $4x + 7y - 3 = 0$ and $2x - 3y + 1 = 0$ that has equal intercepts on the axes.

The equation of a line passing through $(p \cos \alpha, p \sin \alpha)$ and making an angle $(90^\circ + \alpha)$ with the positive direction of the $X$-axis is:

$A$ straight line through the point $P(1, 2)$ makes an angle $\theta$ with the positive $X$-axis in the anticlockwise direction and meets the line $x + \sqrt{3}y - 2\sqrt{3} = 0$ at $Q$. If $PQ = \frac{1}{2}$,then $\theta =$