Exercise · ZW-OPEN-BALL-008 · ★★☆ Intermediate
How many points \((a,b)\) with integer coordinates lie in the Euclidean ball \(B((0,0),3)\)?
Choose one option.
Show answer
- B. 25
Explanation
Count the integers with \(a^2+b^2<9\): for each \(a\in\{-2,\dots,2\}\) there are 5 values of \(b\), and \(a=\pm3\) fails. Total: 25 (the 4 points with \(a^2+b^2=9\) are outside).
Review the concept: Open ball
The open ball \(B(x,r)\) with center \(x\) and radius \(r>0\) in a metric space \((X,d)\) is the set of points of \(X\) whose distance to \(x\) is less than \(r\).
\[B(x,r)=\{\,y\in X : d(x,y)
More practice problems: open ball
- ZW-OPEN-BALL-001 In ℝ with d(x,y)=|x-y|, which interval is the open ball B(2,3)?
- ZW-OPEN-BALL-002 In the usual ℝ, which of these points belongs to B(0,2)?
- ZW-OPEN-BALL-003 In Euclidean ℝ^2, which point lies in B((0,0),1)?
- ZW-OPEN-BALL-004 With the max distance d_∞ on ℝ^2, the ball B((0,0),2) is a square. What is its area?
- ZW-OPEN-BALL-005 With the taxicab distance d_1 on ℝ^2, the ball B((0,0),1) is a rhombus with vertices (pm1,0) and (0…
- ZW-OPEN-BALL-006 In a space with the discrete metric, what is the ball B(a,1)?
- ZW-OPEN-BALL-007 In the usual ℝ, what is the intersection B(0,2)∩ B(3,2)?
- ZW-OPEN-BALL-009 Let y∈ B(a,r) and s=r-d(a,y). In ℝ with a=0, r=5 and y=3, what is s, the radius ensuring B(y,s)⊆ B(…