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AMO 2022 – Junior College (Grade 11–12) Question 9

AMO · AMO 2022 – Junior College (Grade 11–12)

Question 9

In the $xz$-plane, segment $AB$ joins $A=(0,0,4)$ to $B=(3,0,0)$. Rotating $AB$ about the $z$-axis sweeps a cone. Which equation describes its surface? Clarification: restrict the selected equation to $0\le z\le4$, because a segment sweeps only a finite surface.

Question illustration
  1. $16x^2+16y^2-9z^2+72z-144=0$
  2. $4x^2+16y^2+z^2+16-25=0$
  3. $3x^2+4y^2-5z^2+25z-1=0$
  4. $x^2+y^2-9z^2+25=0$
  5. $x^2+y^2+z^2-1=0$