The equation of the locus of the point of intersection of the straight lines $$x\sin \theta + (1 - \cos \theta )y = a\sin \theta $$ and $$x\sin \theta - (1 + \cos \theta )y + a\sin \theta = 0$$ is

Consider three points $P(\cos \alpha, \sin \beta), Q(\sin \alpha, \cos \beta)$ and $R(0,0)$, where $0<\alpha, \beta<\frac{\pi}{4}$. Then

The line parallel to the $x$-axis passing through the intersection of the lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$ where $(a, b) \neq(0,0)$ is

If $$(1,5)$$ be the midpoint of the segment of a line between the line $$5 x-y-4=0$$ and $$3 x+4 y-4=0$$, then the equation of the line will be