Find the value of $k$ for which the line $(k-3) x - (4-k^2) y + k^2 - 7k + 6 = 0$ passes through the origin.

The intercept form of the equation of the straight line passing through the point $(4, -3)$ and perpendicular to the line passing through the points $(1, 1)$ and $(2, 3)$ is

$A$ line passes through $(x_{1}, y_{1})$ and $(h, k)$. If the slope of the line is $m$,show that $k - y_{1} = m(h - x_{1})$.

The equations of the lines which make intercepts on the axes whose sum is $8$ and product is $15$ are

The intercept of a line between the coordinate axes is divided by the point $(-5, 4)$ in the ratio $1 : 2$. The equation of the line will be:

$A$ line passing through the point $P(1, 1)$ and parallel to the line $x - y = 5$ cuts the line $x + 3y - 2 = 0$ at $Q$. Then twice the length of the segment $PQ$ is