Project ForgeEngineering Mathematics
Calculus
GATE 2009
Hard
The distance between the origin and the point nearest to it on the surface z²=1+xy is
- 1
- B√3/2
- C√3
- D2
Explanation
Minimize x²+y²+z² subject to z²-xy=1. Lagrange multipliers give a critical point at x=y=0, z=±1 (distance 1). The boundary case z=0 (xy=-1) gives minimum x²+y²=2 via AM-GM (distance √2) by comparison, which is larger. So the true nearest point is at distance 1.
Source: GATE ME 2009 (2009)
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