Find the equation of the line passing through the point $(2, 3)$ and making an angle of $\frac{\pi}{4}$ with the line $2x + 3y = 5$.

Find the angle between the lines joining the points $(0, 0), (2, 3)$ and $(2, -2), (3, 5)$.

The acute angle between the lines $x \cos 30^{\circ} + y \sin 30^{\circ} = 3$ and $x \cos 60^{\circ} + y \sin 60^{\circ} = 5$ is (in $^{\circ}$)

If the straight lines $2x + 3y - 3 = 0$ and $x + ky + 7 = 0$ are perpendicular,then the value of $k$ is:

Consider a triangle $ABC$ having the vertices $A(1,2)$,$B(\alpha, \beta)$,and $C(\gamma, \delta)$. The angles are $\angle ABC = \frac{\pi}{6}$ and $\angle BAC = \frac{2\pi}{3}$. If the points $B$ and $C$ lie on the line $y = x + 4$,then $\alpha^2 + \gamma^2$ is equal to:

If the lines $y = 3x + 1$ and $2y = x + 3$ make equal angles with the line $y = mx + 4$,then $m = $ . . .