The foot of the perpendicular drawn from $(2, 4)$ to the line $x + y = 1$ is

If $(h, k)$ is the image of the point $(3, -4)$ with respect to the line $2x - 3y - 5 = 0$ and $(\ell, m)$ is the foot of the perpendicular from $(h, k)$ onto the line $3x + 2y + 12 = 0$,then $\ell h + mk + 1 =$ ?

$A$ person standing at the junction (crossing) of $2$ straight paths represented by the equations $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$,wants to reach the path whose equation is $6x - 7y + 8 = 0$ in the least time. The equation of the path he should follow 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

$A$ ray of light passing through the point $(2, 3)$ reflects on the $Y$-axis at a point $P$. If the reflected ray passes through the point $(3, 2)$ and $P = (a, b)$,then $5b =$

If a ray travels along the line $x = 1$ and gets reflected by the line $x + y = 1$,find the equation of the reflected ray.