Find the separate equations of the straight lines represented by $2x^2 + 7xy + 3y^2 = 0$, and find the acute angle $\theta$ between them.

Prove that the equation $x^2 + 6xy + 9y^2 + 4x + 12y - 5 = 0$ represents a pair of parallel straight lines, and find the distance between them.

Find the value of $k$ so that the equation $$2x^2 + 5xy + 2y^2 + 7x + 5y + k = 0$$ represents a pair of straight lines. For this value of $k$, find their point of intersection.

Prove that the homogeneous equation of second degree $ax^2 + 2hxy + by^2 = 0$ represents two straight lines passing through the origin. Hence, prove that the angle $\theta$ between the two lines is given by $\tan\theta = \pm \frac{2\sqrt{h^2 - ab}}{a + b}$, and state the conditions for perpendicularity and coincidence.