Regions of the plane defined by inequalities

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Consider the two point sets $A = \{(x;y)\mid y^2 -x^2 -2x \ge 10\}, \qquad B = \{(x;y)\mid 2x + y \le 4\}.$

• (a) represent $A\cap B$ (solid or dashed boundary as appropriate);
• (b) represent $B - A$ (set difference);
• (c) decide whether $P(0;0)$, $Q(-2;1)$, $R(3;-3)$ belong to $A\cap B$.

(The opening part, with three figures to interpret, is omitted as it depends on images.)

Solution

The boundary of $A$ is $y^2-x^2-2x=10$, i.e. $y^2-(x+1)^2=9$: a vertical hyperbola with centre $(-1;0)$, $a=3$. $A$ is the region “inside” the branches (where $y^2-(x+1)^2\ge9$); solid boundary ($\ge$). $B$ is the half-plane $2x+y\le4$, solid boundary.

(a) $A\cap B$: points both inside the hyperbola and on/below the line $2x+y=4$.

(b) $B-A$: points of the half-plane $B$ not in $A$ (outside the hyperbola).

(c) Check $A$‘s condition $y^2-x^2-2x\ge10$:

• $P(0;0)$: $0\ge10$? No → $P\notin A$.
• $Q(-2;1)$: $1-4+4=1\ge10$? No → $Q\notin A$.
• $R(3;-3)$: $9-9-6=-6\ge10$? No → $R\notin A$.

None of the three is in $A$, so none belongs to $A\cap B$ (although all lie in $B$).

$\boxed{P,Q,R\notin A\cap B}$