GATE 2022 ST – Question 63
A point P=(x,y,z) lies on z=xy+5 and x+y+z=1 and is nearest to the origin. Find its distance from the origin as an integer.
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Correct answer: 3 to 3
Explanation
Put s=x+y and p=xy. Then z=1−s and p=−s−4. Real x,y require s²−4p=(s+2)²+12>0,so every s is feasible. Squared distance=x²+y²+z²=s²−2p+(1−s)²=2s²+9. The minimum is 9 at s=0,with p=−4,giving (x,y,z)=(2,−2,1) or (−2,2,1). Distance **3**.