Let A=(x,y) R R 2x^2+2y^2-2x-2y=1, B=(x,y) R R 4x^2+4y^2-16y+7=0, C=(x,y) R R x^2+y^2-4x-2…

Question

Let $A=\{(x,y)\in\mathbb R	imes\mathbb R\mid 2x^2+2y^2-2x-2y=1\}$, $B=\{(x,y)\in\mathbb R	imes\mathbb R\mid 4x^2+4y^2-16y+7=0\}$, $C=\{(x,y)\in\mathbb R	imes\mathbb R\mid x^2+y^2-4x-2y+5\le r^2\}$. Then the minimum value of $|r|$ such that $A\cup B\subseteq C$ is equal to

Accepted Answer

$A$: $(x-\tfrac12)^2+(y-\tfrac12)^2=1$, centre $(\tfrac12,\tfrac12)$, radius $1$.
$B$: $x^2+(y-2)^2=\tfrac94$, centre $(0,2)$, radius $\tfrac32$.
$C$: $(x-2)^2+(y-1)^2\le r^2$, centre $(2,1)$.

For $A\subset C$: $r\ge\sqrt{(2-\frac12)^2+(1-\frac12)^2}+1=\sqrt{\frac52}+1$.

For $B\subset C$: $r\ge\sqrt{(2-0)^2+(1-2)^2}+\frac32=\sqrt5+\frac32$.

The binding (larger) requirement is from $B$:
$$r=\sqrt5+\frac32=\frac{3+2\sqrt5}2.$$

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