Previous Year Question
The intersection of the two surfaces, ρ = 2 and z = 1, is
Correct answer: B
Explanation
In cylindrical coordinates, ρ = 2 is a right circular cylinder of radius 2 about the z-axis, while z = 1 is a horizontal plane. Their intersection is x² + y² = 4 at z = 1, which is a circle.
Formula
ρ=2 ⇒ x²+y²=4; z=1 ⇒ circle at z=1
Step-by-step solution
Correct Answer
B — a circle
Concept and Reasoning
In cylindrical coordinates, ρ = 2 is a right circular cylinder of radius 2 about the z-axis, while z = 1 is a horizontal plane. Their intersection is x² + y² = 4 at z = 1, which is a circle.
Calculation / Key Relation
ρ=2 ⇒ x²+y²=4
z=1
Intersection: x²+y²=4 at z=1 ⇒ circle
Exam Takeaway
Cylinder ∩ horizontal plane = circle.
Common Mistake
Do not interpret ρ=2 as a sphere; spherical radial distance is normally denoted r.
Common mistake
Do not interpret ρ=2 as a sphere; spherical radial distance is normally denoted r.
Exam tip
Cylinder ∩ horizontal plane = circle.
Quick method
Cylinder ∩ horizontal plane = circle.