Suppose the pairs of straight lines $x^2 - 2axy - y^2 = 0$ and $x^2 - 2bxy - y^2 = 0$ are such that each pair bisects the angles between the other. Then $ab =$

The equation of a pair of lines $y=px$ and $y=qx$ can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of angle bisectors of the lines $x^2-4xy-5y^2=0$ is

The combined equation of the bisectors of the angle between the lines represented by $({x^2} + {y^2})\sqrt{3} = 4xy$ is

The joint equation of the pair of lines passing through the point of intersection of the lines represented by $2x^{2}-xy-15y^{2}-7x+32y-9=0$ and parallel to the coordinate axes is:

The area (in sq. units) of the triangle formed by the straight line $x+y=3$ and the angular bisectors of the pair of straight lines $x^2-y^2+2y=1$ is

If the lines represented by $x^2-2hxy-y^2=0$ are rotated about $(0,0)$ through an angle $\alpha$,one in the clockwise direction and the other in the counter-clockwise direction,then the combined equation of the bisectors of the angle between the lines thus obtained is