The area of the region bounded by the locus of point P satisfying d(P,A)=4, where A is (1,2) is _______ .
Where we define the distance between two points P(x,y) and Q(a,b) as $$d(P,Q)=max(|a-x|,|b-y|)$$.
My attempt
$$d(P,A)=max(|1-x|,|2-y|)$$ $$4=max(|1-x|,|2-y|)$$ Now it gives 4 cases to be equaled to 4 which gives different coordinates.
Then how will I know which coordinate to take?
Maximum of two numbers is $4$ iff one of them is $4$ and the other one is $\leq 4$. From this you can check that the locus consists of points on the rectangle with vertices $(-3,-2),(-3,6),(5,-2)$ and $(5,6)$. [For example, $|1-x|=4$ iff $x=-3$ or $x=5$]. The area of this rectangular region is 64.