$A$ line passes through the origin and is perpendicular to two given lines $2x + y + 6 = 0$ and $4x + 2y - 9 = 0$. What is the ratio in which the origin divides the segment formed by the intersection points of this line with the two given lines?

The least distance from the origin to a point on the line $y=x+3$ which lies at a distance of $2$ units from $(0,3)$ is

Consider the family of lines $x(a + b) + y = 1$,where $a, b$ and $c$ are the roots of the equation $x^3 - 3x^2 + x + \lambda = 0$ such that $c \in [1, 2]$. If the given family of lines makes a triangle of area $A$ with the coordinate axes,then the maximum value of $A$ (in sq. units) will be:

If $(1, 5)$ is the midpoint of the segment of a line between the lines $5x - y - 4 = 0$ and $3x + 4y - 4 = 0$,then the equation of the line is:

Let the lines $3x - 4y - \alpha = 0$,$8x - 11y - 33 = 0$,and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$,then $|\alpha \lambda|$ is equal to:

If a line $ax + 2y = k$ forms a triangle of area $3$ sq. units with the coordinate axes and is perpendicular to the line $2x - 3y + 7 = 0$,then the product of all the possible values of $k$ is