The equation of the line passing through $(4, -6)$ and making an angle of $45^\circ$ with the positive $x$-axis is:

Three points $P(h, k)$,$Q(x_{1}, y_{1})$ and $R(x_{2}, y_{2})$ lie on a line. Show that $(h-x_{1})(y_{2}-y_{1}) = (k-y_{1})(x_{2}-x_{1})$.

Two lines $L_1$ and $L_2$ passing through the point $P(1, 2)$ cut the line $x+y=4$ at a distance of $\frac{\sqrt{6}}{3}$ units from $P$. Then the angles made by $L_1$ and $L_2$ with the positive $X$-axis are

If the point $(5, 2)$ divides the segment of the line between the axes into two equal parts,then its equation is:

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

Find the equation of a line passing through the point $(2, 5/3)$ such that the segment of the line intercepted between the axes is divided by the point in the ratio $5 : 4$.