Question

The area of the region bounded by the locus of a point $P$ satisfying $d(P,A)=4$, where $A$ is $(1,2)$ is

• A. $64sq$. unit
• B. $54sq$. unit
• C. $16\pi$ sq. unit
• D. None of these

Answer 1

Answer: (A)

We have, $\max \{|x-1|,|y-2|\}=4$
If $\{|x-1|\ge |y-2|\}$
then $|x-1|=4$,
i.e., if $(x+y-3)(x-y+1)\ge 0$,
then $x=-3$ or 5
If $|y-2|\ge |x-1|$,
then $|y-2|=4$
i.e., $(x+y-3)(x-y+1)\le 0$,
then $y=-2$ or 6 .
So, the locus of $P$ bounds a square, the equation of whose sides are
$x=-3,x=5,y=-2,y=6$
Thus, the area is $(8{)}^2=64Sq$. unit.