If one of the lines given by the pair of lines $3 x^2-2 y^2+a x y=0$ is making an angle $60^{\circ}$ with $X$-axis, then $a=$

$A$ straight line passing through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at the points $P$ and $Q$ respectively. Then, the point $O$ divides the line segment $P Q$ in the ratio

If the axes are translated to the orthocentre of the triangle formed by the points $\mathrm{A}(7,5), \mathrm{B}(-5,-7)$ and $C(7,-7)$, then the coordinates of the incentre of the triangle in the new system are

The angle made by a line $L$ with positive $X$-axis measured in the positive direction is $\frac{\pi}{6}$ and the intercept made by $L$ on $Y$-axis is negative. IF $L$ is at a distance of 5 units from the origin, then the perpendicular distance from the point $(1,-\sqrt{3})$ to the line $L$ is