The line $y = 2x + c$ passes through a point that is equidistant from both the axes and lies in the first quadrant $(x > 0, y > 0)$. Then the value of $c$ is...

The equation of the straight line passing through the point $(3, 2)$ and perpendicular to the line $y = x$ is

The equation of the line passing through the origin and the point $(a \cos \theta, a \sin \theta)$ is .....

The line $L_1$ passes through the points $(-1, 2)$ and $(3, 6)$. If $L_1$ divides the line $L_2$ (which passes through $(3, -1)$) in the ratio $1:3$ internally, and the point of intersection lies on $L_1$, find the equation of $L_2$ given that it is perpendicular to $L_1$.

If three points $(h, 0), (a, b),$ and $(0, k)$ lie on a line,show that $\frac{a}{h} + \frac{b}{k} = 1$.

Find the value of $k$ for which the line $(k-3)x - (4-k^2)y + k^2 - 7k + 6 = 0$ is parallel to the $x$-axis.