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

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

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