The joint equation of pair of lines through the origin and making an equilateral triangle with the line $$y = 5$$ is

The equations of the lines passing through the point $$(3,2)$$ and making an acute angle of $$45^{\circ}$$ with the line $$x-2 y-3=0$$ are

If the sum of slopes of lines represented by $$\mathrm{ax^2+8xy+5y^2=0}$$ is twice their product, then a =

If the line joining two points $$\mathrm{A}(2,0)$$ and $$\mathrm{B}(3,1)$$ is rotated about $$\mathrm{A}$$ in anticlockwise direction through an angle of $$15^{\circ}$$, then the equation of the line in new position is