Let $P$ be the point of intersection of the lines $L_1 \equiv x-y-7=0$ and $L_2 \equiv x+y-5=0 . A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ are points on the lines $L_1=0$ and $L_2=0$ respectively such that $P A=3 \sqrt{2}$, $P B=\sqrt{2}, x_1, y_1 \geq 0, x_2, y_2 \geq 0$, then the angle made by the line segment $A B$ at the origin is

Let $A(2,1)$ be a point and equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $C$ is distance of the point $A$ from origin such that $a=b c$, then locus of $P$ is

The point $(4,1)$ undergoes the following transformations successively :

(i) Reflection is the line $x-y=0$

(ii) Shifting through a distance of 2 units along the positive $X$-axis

(iii) Projection on $X$-axis

The coordinates of the point in its final position are

Two straight lines $3 x+4 y=5$ and $4 x-3 y=15$ intersect at the point $A$. The equations of the lines passing through $(1,2)$ and intersecting the given lines at $B$ and $C$ such that $A B=A C$ are