The equation of a straight line passing through $(-3, 2)$ and cutting an intercept equal in magnitude but opposite in sign from the axes is given by:

$\beta$ is the angle made by the perpendicular drawn from the origin to the line $L \equiv x+y-2=0$ with the positive $X$-axis in the anticlockwise direction. If '$a$' is the $X$-intercept of the line $L=0$ and $p$ is the perpendicular distance from the origin to the line $L=0$,then $a \tan \beta + p^2 =$

The equation of a line with slope $-\frac{1}{\sqrt{2}}$ and making an intercept of $2 \sqrt{2}$ units on the negative direction of the $y$-axis is:

Find the value of $x$ for which the points $(x, -1), (2, 1),$ and $(4, 5)$ are collinear.

$P(6,4)$ is a point on the line $x-y-2=0$. If $A(\alpha, \beta)$ and $B(\gamma, \delta)$ are two points on this line lying on either side of $P$ at a distance of $4$ units from $P$,then $\alpha^2+\beta^2+\gamma^2+\delta^2=$

The number of integral values of $m$ such that the abscissa of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer,is