If $B$ is the reflection of the point $A(1, 2)$ with respect to the line $y = x$ and $(\alpha, \beta)$ is the reflection of $B$ with respect to $y = 0$,then-

The equation of the straight line passing through the point of intersection of the lines $5x - 6y - 1 = 0$ and $3x + 2y + 5 = 0$ and perpendicular to the line $3x - 5y + 11 = 0$ is

If $Q$ is the image of the point $P(1,1)$ with respect to the straight line $x+y+1=0$,then the length of the perpendicular drawn from $Q$ to the line $3x-4y+3=0$ is

Let $0 < \alpha < \pi/2$ be a constant angle. If $P \equiv (\cos \theta, \sin \theta)$ and $Q \equiv (\cos(\alpha - \theta), \sin(\alpha - \theta))$,how is $Q$ obtained from $P$?

$A$ 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:

The unit normal vector to the line joining $i - j$ and $2i + 3j$ and pointing towards the origin is