Let A = {(α, β) ∈ R × R : ∣α − 1∣ ≤ 4 and ∣β − 5∣ ≤ 6} and B = {(α, β) ∈ R × R : 16(α − 2)² + 9(β − 6)² ≤ 144}. Then:
❓ Question Let A = { ( α , β ) ∈ R × R : ∣ α − 1 ∣ ≤ 4 , ∣ β − 5 ∣ ≤ 6 } and B = { ( α , β ) ∈ R × R : 16 ( α − 2 )² + 9 ( β − 6 )² ≤ 144 } . Then determine the relationship between sets A and B (containment, intersection, union, etc.). 🖼️ Question Image ✍️ Short Solution Step 1 — Convert descriptions into ranges/standard form For A A A : ∣ α − 1 ∣ ≤ 4 ⇒ α ∈ [ − 3 , 5 ] |\alpha-1|\le4 \Rightarrow \alpha\in[-3,5] ∣ β − 5 ∣ ≤ 6 ⇒ β ∈ [ − 1 , 11 ] |\beta-5|\le6 \Rightarrow \beta\in[-1,11] So A A is an axis-aligned rectangle with opposite corners at ( − 3 , − 1 ) (-3,-1) and ( 5 , 11 ) (5,11) . Center at ( 1 , 5 ) (1,5) , width = 8 =8 , height = 12 =12 . For B B : divide the inequality by 144 144 : ( α − 2 ) 2 9 + ( β − 6 ) 2 16 ≤ 1. So B B is an ellipse centered at ( 2 , 6 ) (2,6) with semi-axes ...