I have a problem in loci

Question

$ABCD$ is a square with side length $4$ cm. A variable point $P$ moves inside the square so that $PA\leq 4$ cm, $PC\leq PA$ and the area of $ABP$ is $\leq 6$ cm$^2$. Construct $ABCD$ accurately and shade the region in which $P$ must lie.

Answer 1

$PA\leq 4$ implies that $P$ lies inside the quarter of circle with centre in $A$ through $B,D$. $PC\leq PA$ implies that $P$ is closer to $C$ than to $A$, so $P$ and $C$ lie on the same side of the diagonal $BD$. At last, $[ABP]\leq 6$ implies that the distance of $P$ from the $AB$-side is at most $3$. The allowed region for $P$ is so the following one (dark blue, closed region):

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