Open Ball that bounds C

Question

For an arbitrary choice of $a,b,r∈\mathbb{R}$, determine the open ball that bounds $C=\left \{(x, y)∈\mathbb{R}^2:(x − a)^2 + (y − b)^2 ≤ r^2\right \}$. How does one approach this question? Is it along the lines of $B_{r}\left ( x,y \right )=...$?

Answer 1

You have to find a center and a radius, such that the ball with that radius and center, contains $C$. Can you find such a center and radius?

To do this, first understand the shape of $C$. $C$, with the Euclidean distance in mind, is in fact a ball itself, with center $(x,y)$ and radius $r$ , since the distance between $(a,b)$ and $(x,y)$ in Euclidean metric is $\sqrt{(x-a)^2 + (y-b)^2}$.

Now, all you need is a ball with the same center $(x,y)$, and a larger radius $r+1$ say, then this ball covers $C$, as can be seen easily.