The line passing through the points $(-2, 6)$ and $(4, 8)$ is perpendicular to the line passing through the points $(8, 12)$ and $(x, 24)$. Find the value of $x$.

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

The equation of the line bisecting the join of $(3, -4)$ and $(5, 2)$ and having its intercepts on the $x$-axis and the $y$-axis in the ratio $2 : 1$ 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

The gradient of the line joining the points on the curve $y = x^2 + 2x$ whose abscissae are $1$ and $3$,is

The length $L$ (in centimetre) of a copper rod is a linear function of its Celsius temperature $C$. In an experiment,if $L = 124.942$ when $C = 20$ and $L = 125.134$ when $C = 110,$ express $L$ in terms of $C$.