Find the value of $x$ for which the points $(x, -1), (2, 1),$ and $(4, 5)$ are collinear.

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

$A$ line passes through $(2, 2)$ and is perpendicular to the line $3x + y = 3$. Its $y$-intercept is

$A$ line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive $x$-axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha}{2}$ in the clockwise direction. If in the new position,the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$,then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is

Find the number of integer values of $m$ such that the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is an integer.

The line passing through $(1, 0)$ and $(-2, \sqrt{3})$ makes an angle of ...... with the $x$-axis. (in $^o$)