If $PM$ is the perpendicular from $P(2, 3)$ onto the line $x + y = 3$,then the coordinates of $M$ are

The locus of the image of a variable point $P(\alpha, 2 \alpha-1)$ with respect to the line $3 x-2 y+4=0$ is

$A$ straight line $L_1$ passing through $A(3,1)$ meets the coordinate axes at $P$ and $Q$ such that its distance from the origin $O$ is maximum. Then the area of $\triangle OPQ$ is (in sq. units):

The value of $a$ for which the image of the point $(a, a - 1)$ with respect to the line mirror $3x + y = 6a$ is the point $(a^2 + 1, a)$ is:

Two straight lines are drawn through the point $(0, 2)$ such that the lengths of the perpendiculars from the point $(4, 4)$ to these lines are each equal to $2$ units. The equation of the line joining the feet of these perpendiculars is

If $2x + 3y = 5$ is the perpendicular bisector of the line segment joining the points $A\left(1, \frac{1}{3}\right)$ and $B$,then $B$ is equal to