Find the two points on the line $x + y = 4$ which are at a unit distance from the line $4x + 3y = 10$.

The equations of two lines passing through $(0, a)$ which are at a distance of $a$ from the point $(2a, 2a)$ are:

For an integer $K$,if the point $P(K^2, K+1)$ and the origin $O(0,0)$ lie in the same region between the lines $x+2y-5=0$ and $3x-y+1=0$,then the possible number of such points $P$ is

Lines are drawn parallel to the line $4x - 3y + 2 = 0$ at a distance of $\frac{3}{5}$ from the origin. Which one of the following points lies on any of these lines?

$A$ line $L_1$ passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y=0$ is parallel to the line $L_2$. If $L_2$ passes through the origin and also through the point of intersection of the lines $3x-y+2=0$ and $x-3y-2=0$,then the distance between the lines $L_1$ and $L_2$ is

Let two points be $A(1, -1)$ and $B(0, 2)$. If a point $P(x', y')$ is such that the area of $\Delta PAB = 5 \; \text{sq units}$ and it lies on the line $3x + y - 4\lambda = 0$,then a value of $\lambda$ is